SPECTRUM OF THE RUELLE OPERATOR AND EXPONENTIAL DECAY OF CORRELATIONS FOR OPEN BILLIARD FLOWS

By Luchezar Stoyanov

Abstract. The paper deals with the billiard flow in the exterior of several strictly convex disjoint domains in the plane with smooth boundaries satisfying an additional (visibility) condition. Using a modification of the technique of Dolgopayt [D], we get spectral estimates for the Ruelle operator related to a Markov family for the non-wandering (trapping) set of the flow similar to these in [D] in the case of transitive Anosov flows on compact manifolds with smooth jointly non-integrable horocycle foliations. As a consequence, we get exponential decay of correlation for Hölder continuous potentials on the non-wandering set. Combining the spectral estimate for the Ruelle operator with an argument of Pollicott and Sharp [PoS], one also derives the existence of a meromorphic continuation of the dynamical zeta function of the billiard flow to a half-plane Re(s) < h_{T} − ϵ, where h_{T} is the topological entropy of the billiard flow, and an asymptotic formula with an error term for the number π(λ) of closed orbits of least period λ > 0.

1 Introduction

Let K be a subset of IR^{2} of the form K = K_{1} ∪ K_{2} ∪ … ∪ K_{k_{0}}, where K_{i} are compact strictly convex disjoint domains in IR^{2} with C^{p} (p ≥ 5) boundaries ∂K_{i} and k_{0} ≥ 3. Set Ω = \overline{IR^{2} \ K}.

We will assume that K satisfies the following condition: for every pair K_{i}, K_{j} of different connected components of K the convex hull of K_{i} ∪ K_{j} has no common points with any other connected component of K. With this condition, the billiard generated in the exterior of K is sometimes called an open billiard.

As usual, we will denote by T^{∗}(Ω) the cotangent bundle of Ω and by S^{∗}(Ω) its cosphere bundle. The billiard flow ϕ_{t} (also known as the broken geodesic flow) is defined on S^{∗}(Ω) (and more generally on T^{∗}(Ω)\{0}) in the standard way. Clearly this flow has singularities, however its restriction to the non-wandering set Λ only has simple discontinuities at reflection points. Moreover, Λ is compact, ϕ_{t} is hyperbolic and transitive on Λ, and it follows from [Mor] that ϕ_{t} is non-lattice and therefore by a result of Bowen [B1], it is topologically weak-mixing on Λ.

Our main result in this paper is a spectral estimate for the Ruelle operator similar to that of Dolgopayt [D] in the case of transitive Anosov flows on compact manifolds with C^{1} jointly non-integrable horocycle foliations. Let R = {R_{i}}_{i=1}^{k} be a Markov family for ϕ_{t} near Λ consisting of rectangles R_{i} = [U_{i} ∩ Λ, S_{i} ∩ Λ], where U_{i} (resp. S_{i}) are local unstable (resp. stable) manifolds near Λ (cf. Sect.2 for details). The first return time function τ : R = ∪_{i=1}^{k}R_{i} ⟶ [0, ∞) and the standard Poincaré map P : R ⟶ R have natural C^{1} extensions on a neighbourhood of R in the union of the corresponding cross-sections containing the rectangles R_{i}. Setting U = ∪_{i=1}^{k}U_{i}, the shift map σ : U ∩ Λ ⟶ U ∩ Λ is defined by σ = p∘P, where p : R ⟶ U ∩Λ is the projection along the leaves of local stable manifolds. It follows from the existence of tubular families for Λ (cf. Robinson [Ro]) that σ has a C^{1} extension on a neighbourhood of U ∩Λ in U (cf. Sect. 2 for details), and choosing the size of the rectangles R_{i} sufficiently small, we may assume that σ has a C^{1} extension σ : U ⟶ U. Given a C^{1} real-valued function f on U, set g = f − P τ, where P ∈ IR is the unique number such that the topological pressure of g with respect to σ is zero (cf. e.g. [PP2]). For a, b ∈ IR, one defines the Ruelle operator L_{g−(a+ib)τ} : C^{1}(U) ⟶ C^{1}(U) in the usual way (cf. Sect.2). Here C^{1}(U) is the space of continuously differentiable complexvalued functions h with ∥dh∥ = sup_{u∈U} ∥dh(u)∥ < ∞. We will denote by ∥h∥ the standard sup norm of h.

Theorem 1.1. For every ϵ > 0 there exist constants ρ ∈ (0, 1), a_{0} ∈ (0, 1), b_{0} ≥ 1 and C > 0 such that for every integer n > 0, every β > 0 and every h ∈ C^{1}(U), if a, b ∈ IR, are such that |a| ≤ a_{0}, |b| ≥ b_{0}, then

|L_{f−(P +a+ib)τ}^{n}h∥_{1,b} ≤ C ρ^{n} |b|^{ϵ} ∥h∥_{1,b} ,

where the norm ∥.∥_{1,b} on C^{1}(U) is defined by ∥h∥_{1,b} = ∥h∥ + \frac{∥dh∥}{|b|} for |b| ≥ 1. In particular, for the spectral radius ρ(L_{f−(P +a+ib)τ}) of L_{f−(P +a+ib)τ} in C^{1}(U) we have ρ(L_{f−(P +a+ib)τ}) ≤ ρ < 1.

Similar estimate holds for the Ruelle operator acting on the space F_{β}(U ∩ Λ) of Hölder continuous functions with respect to an appropriate norm ∥ · ∥_{β,b,U∩Λ}; see Corollary 3.3 below for details.

Theorem 1.1 is obtained as a consequence of the central Theorem 3.1 which in turn is analogous to Lemma 4 in [D]. As another consequence of Theorem 3.1 and the procedure described in [D] (based on previous arguments of Pollicott [Po] and Ruelle [R2]) one gets exponential decay of correlations for open billiard flows in the plane.

Throughout, given α > 0, we denote by F_{α}(Λ) the set of Hölder continuous functions with Hölder exponent α and by ∥h∥_{α} the Hölder constant of h ∈ F_{α}(Λ).

Theorem 1.2. Let F be a Hölder continuous function on Λ in S^{∗}(Ω) and let ν_{F} be the Gibbs measure determined by F on Λ. For every α > 0 there exist constants C = C(α) > 0 and c = c(α) > 0 such that

|∫_{Λ} A(x)B(ϕ_{t}(x)) dν_{F} (x) − (∫_{Λ} A(x) dν_{F} (x)) (∫_{Λ} B(x) dν_{F} (x))| ≤ Ce^{−ct}∥A∥_{α} ∥B∥_{α}

for any two functions A, B ∈ F_{α}(Λ).

Using Theorem 1.1 and applying the argument of Pollicott and Sharp [PoS] in the case of geodesic flows on compact surfaces of negative curvature, one derives that the zeta function

ζ(s) = ∏_{γ}(1 − e^{−sℓ(γ)})^{−1}

of the billiard flow ϕ_{t} has an analytic continuation in a half-plane Re(s) > c_{0} for some c_{0} < h_{T} except for a simple pole at s = h_{T}. Here h_{T} is the topological entropy of the flow ϕ_{t}, γ denotes a closed orbit of ϕ_{t} and ℓ(γ) is the least period of γ. Moreover, folowing [PoS] again (cf. also [DP]), one derives that there exists c ∈ (0, h_{T}) such that

π(λ) = #{γ : ℓ(γ) ≤ λ} = li(e^{h_{T} λ}) + O(e^{cλ})

as λ → ∞, where li(x) = ∫_{2}^{x} du/ log u. The latter is a much stronger result than the standard Prime Orbit Theorem for open planar billiards in [Mor] (cf. [St1] for the higher dimensional case) derived by means of a result of Parry and Pollicott [PP1].

The present work was motivated by Ikawa’s approach ([I2]) to the study of the so called Modified Lax-Phillips Conjecture (cf. the Epilogue in [LP] or Sect. 4 in [Z]) concerning the distribution of poles of the scattering matrix in obstacle scattering for the wave equation with Dirichlet boundary condition. In the case when the obstacle K in IR^{n} (n ≥ 3, n odd) is a finite disjoint union of strictly convex bodies Ikawa showed that the MLPC is equivalent to the existence of analytic singularities of a specially defined zeta function ζ_{D}(s). As for the classical dynamical zeta function ζ(s), the Ruelle operator related to the billiard flow on S^{∗}(Ω) turns out to be an useful tool for the study of the scattering zeta function ζ_{D}(s) as well (cf. [I2], [P]). Using such a technique, Ikawa proved the MLPC for special classes of obstacles^{1} in IR^{3}. It is natural to expect that estimates as that in Theorem 1.1 in the higher dimensional case would lead to the proof of the MLPC for more general classes of obstacles.

What concerns results on various types of decay of correlations, the reader is referred to the survey article of Baladi [Ba] (cf. also the Addendum by Dolgopyat and Pollicott [DP]) for general information and historical remarks. Amongst the most recent achievements one should mention the important articles of Liverani [L], Young [Y], Chernov [Ch1] and Dolgopyat [D] answering long standing questions. It apppears that for billiards the only results of this type that have been known so far concern the corresponding discrete dynamical system (generated by the billiard ball map from boundary to boundary). To my knowledge, these results are: the subexponential decay of correlations established for a very large class of dispersing billiards by Bunimovich, Sinai and Chernov [BSC] and the exponential decay of correlations for some classes of dispersing billiards in the plane and on the two-dimensional torus established by Young [Y] and Chernov [Ch2] as consequences of their more general arguments. Theorem 1.2 above describes a non-trivial class of billiard flows with exponential decay of correlations for any Hölder continuous potential. The main feature of these flows is that they have a unique maximal compact invariant set which is hyperbolic and has a zero-dimensional intersection with any (un)stable manifold. However they have other special properties that have been essentially used below. It is plausible that Dolgopyat’s method could be used in more general situations than the one considered here.

The paper is organized as follows. Sect. 2 contains some basic definitions and notation along with some preliminary information. In Sect. 3 we state the central result (Theorem 3.1) and the main lemma (Lemma 3.4) needed for its proof. Theorem 1.1 is then derived in the same way as in [D]. Sections 4-8 are devoted to the proof of Lemma 3.4. We follow the method of Dolgopayt in Sects. 6-8 of [D], however essential new difficulties have to be overcome. These are due to the non-compactness of the phase space and the singularity of the Gibbs measures^{2}. As remarked in [DP], one of the main features of the flow required in order to apply Dolgopyat’s method is the Federer property of the corresponding Gibbs measures. Dislike the case of compact Anosov manifolds, this property is non-trivial in the situation considered here. It is proved in Sect. 5 using the distortion bounds in Sect. 4 concerning the discretization map σ : U ⟶ U. These distortion bounds and the uniform non-integrability of the so called temporal distance function established in Sect. 6 (by adapting a result of Katok and Burns [KB]) are later used in Sect. 7. There we construct for every ϵ > 0 a partition of U into closed intervals of length ≤ ϵ such that the ratio of the lengths of any two of these intervals is bounded from above and below by constants independent of ϵ, and such that the so called triple intersection property with respect to Λ is satisfied^{3}: every interval I from the partition that contains a point of Λ is a member of a family of three consecutive intervals from the partition having common points with Λ and lying in the same connected component of U. This construction enables us to exploit the smoothness of σ and use the mean value theorem and similar ”infinitesimal” tools on (or rather near) the Cantor set U ∩ Λ. This is done in Sect. 7, where the main part of Dolgopyat’s method is described and used. In Sect. 8 we very briefly outline how Theorem 1.2 is obtained from the main estimate (more precisely from Corollary 3.3 below) – now the procedure used in [D] applies automatically.

1E.g. when K is a finite union of disjoint balls of the same radius the distaces between them being large compared to the radius.

2U′ ∩ Λ has Lebesgue measure zero in U′ for any local unstable manifold U′.

Acknowledgements. Most of this paper has been written during my visits to the University of Bordeaux I and the Max-Planck-Institute für Mathematik (Bonn). Thanks are due to these institutions for their support and personally to Vesselin Petkov for many useful discussions.

2 Preliminaries

The non-wandering set Λ for the flow ϕ_{t} is the set of those x ∈ S^{∗}(Ω) such that the trajectory {ϕ_{t}(x) : t ∈ IR} is bounded. Notice that the natural projection of ϕ_{t} on the quotient space S^{∗}(Ω)/ ∼, where ∼ is the equivalence relation (q, v) ∼ (p, w) iff q = p and v = w or q = p ∈ ∂K and v and w are symmetric with respect to T_{q}(∂K), is continuous. Moreover whenever both x and ϕ_{t}(x) are in the interior of S^{∗}(Ω) and sufficiently close to Λ, the map y ↦ ϕ_{t}(y) is smooth on a neighbourhood of x. It follows from results of Sinai ([Si1]) that Λ is a hyperbolic set for ϕ_{t}, and it is easily seen that there exists an open neighbourhood U of Λ in S^{∗}(Ω) such that Λ = ∩_{t∈IR}ϕ_{t}(U). In fact, Λ is clearly the maximal compact ϕ_{t}-invariant subset of S^{∗}(Ω). Moreover, it follows from the natural symbolic coding for the natural section of the flow (the so called billiard ball map; cf. Sect.

4) that the periodic points are dense in Λ (cf. e.g. [Mor]), and ϕ_{t} is transitive on λ. Thus, Λ is a basic set for ϕ_{t} and the classical theory of hyperbolic flows applies in the case under consideration (see e.g. Part 4 in the monograph of Katok and Hasselblatt [KH]).

Consider the metric ρ on S^{∗}(IR^{2}) given by ρ((q, v), (p, w)) = ∥q − p∥ + ∥v − w∥. It is easy to see that there exists a metric ρ̃ on S^{∗}(Ω)/ ∼ which generates the topology of the quotient space and coincides with ρ for pairs of points that are not very close to S_{∂K}^{∗}(Ω). The particular choice of this metric will not be important below (in fact we will only need it sifficiently far away from S_{∂K}^{∗}(Ω)), so we are not going to specify it.

For x ∈ Λ \ S_{∂K}^{∗}(Ω) and a sufficiently small ϵ > 0 let

W_{ϵ}^{s}(x) = {y ∈ S^{∗}(Ω) : ρ(ϕ_{t}(x), ϕ_{t}(y)) ≤ ϵ for all t ≥ 0 , ρ(ϕ_{t}(x), ϕ_{t}(y)) →_{t→∞} 0 } ,

3It is enough to know that there exists m ≥ 3, possibly depending on the choice of the partition, such that for any choice of m consecutive intervals of the partition lying in the same connected component of U whose union has a common point with Λ, at least 3 of these intervals have common points with Λ.

W_{ϵ}^{u}(x) = {y ∈ S^{∗}(Ω) : ρ(ϕ_{t}(x), ϕ_{t}(y)) ≤ ϵ for all t ≤ 0 , ρ(ϕ_{t}(x), ϕ_{t}(y)) →_{t→−∞} 0 }

be the (strong) stable and unstable sets of size ϵ. It is easy to show (either by using arguments from [Si2] or directly) that for every x ∈ Λ \ S_{∂K}^{∗}(Ω) and every sufficiently small ϵ > 0, W_{ϵ}^{s}(x) and W_{ϵ}^{u}(x) are 1-dimensional submanifolds of S^{∗}(Ω) such that E^{s}(x) = T_{x}(W_{ϵ}^{s}(x)) ≠ E^{u}(x) = T_{x}(W_{ϵ}^{u}(x)). Notice that both W_{ϵ}^{s}(x) ∩ Λ and W_{ϵ}^{u}(x) ∩ Λ are Cantor sets, i.e. they are infinite compact totally disconnected sets without isolated points.

For A ⊂ S^{∗}(Ω) and a subset I of IR, following [B2] denote ϕ_{I}(A) = { ϕ_{t}(y) : y ∈ A, t ∈ I }. Given ϵ > 0, set

S_{ϵ}^{∗}(Ω) = {x = (q, v) ∈ S^{∗}(Ω) : dist(q, ∂K) > ϵ} , Λ_{ϵ} = Λ ∩ S_{ϵ}^{∗}(Ω) .

In what follows in order to avoid ambiguity and unnecessary complications we will consider stable and unstable manifolds only for points x in S_{ϵ}^{∗}(Ω) or Λ_{ϵ}; this will be enough for our purposes.

It follows from the genaral theory of horocycle foliations (cf. [S] or [KH]) that if ϵ > 0 is sufficiently small, there exists δ > 0 such that if x, y ∈ Λ and ρ(x, y) < δ, then W_{ϵ}^{s}(x)∩Λ and ϕ_{[−ϵ,ϵ]}(W_{ϵ}^{u}(y) ∩ Λ) intersect at exactly one point [x, y]. That is, there exists a unique t ∈ [−ϵ, ϵ] such that ϕ_{t}([x, y]) ∈ W_{ϵ}^{u}(x). Setting Δ(x, y) = t, defines the so called temporary distance function Δ ([Ch1]) which will be used in Sect. 6 below. If z ∈ Λ_{ϵ} and U ⊂ W_{ϵ}^{u}(z) and S ⊂ W_{ϵ}^{s}(z) are closed (i.e. containing their end points) curves containing z and such that U ∩ Λ and S ∩ Λ have no isolated points, then

Π = [U ∩ Λ, S ∩ Λ] = {[x, y] : x ∈ U ∩ Λ , y ∈ S ∩ Λ }

is called a rectangle in Λ_{ϵ}. Notice that Π is ”foliated” by leaves [x, S∩Λ] of stable manifolds.

Let R = {R_{i}}_{i=1}^{k} be a family of rectangles such that each R_{i} is contained in a C^{1} crosssection D_{i} ⊂ S_{ϵ}^{∗}(Ω) to the flow ϕ_{t}. Thus, for each i, R_{i} = [U_{i} ∩ Λ, S_{i} ∩ Λ], where U_{i} and S_{i} are closed curves in W_{ϵ}^{u}(z_{i}) and W_{ϵ}^{s}(z_{i}), respectively, for some z_{i} ∈ Λ_{ϵ}. Set

R = ⋃_{i=1}^{k} R_{i} .

The family R is called complete if there exists T > 0 such that for every x ∈ Λ there exist t_{1} ∈ [−T, 0) and t_{2} ∈ (0, T] with ϕ_{t_{1}}(x) ∈ R and ϕ_{t_{2}}(x) ∈ R. Thus, τ(x) = t_{2}(x) > 0 is the smallest positive time with P(x) = ϕ_{τ(x)}(x) ∈ R, and P : R ⟶ R is the Poincaré map related to the family R. The function τ : R ⟶ [0, ∞) is called the root function (or the first return time) associated with R.

Following [B2] and [Ra] we will say that a complete family R = {R_{i}}_{i=1}^{k} of rectangles in S_{ϵ}^{∗}(Ω) is a Markov family of size χ ∈ (0, ϵ/2) for the billiard flow ϕ_{t} if:

(a) R_{i} ∩ R_{j} = ∅ for i ≠ j and for each i the sets U_{i} ∩ Λ and S_{i} ∩ Λ are contained in the interior of the curves U_{i} and S_{i}, respectively;

(b) diam(R_{i}) ≤ χ ;

(c) For any i ≠ j and any x ∈ R_{i} ∩ P^{−1}(R_{j}) we have P([x, S_{i} ∩ Λ]) ⊂ [P(x), S_{j}] and P([U_{i} ∩ Λ, x]) ⊃ [U_{j}, P(x)];

(d) For any i = 1, …, k and x ∈ R_{i} the function τ is constant on the set [x, S_{i} ∩ Λ].

The existence of a Markov family R of an arbitrarily small size χ > 0 for ϕ_{t} follows from the construction of Bowen [B2] (cf. also Ratner [Ra]). Apart from that, one can easily arrange one extra property of R. Namely, for i ≠ j, (1 ≤ i, j ≤ k_{0}), denote by Π_{i,j} the set of those x = (q, v) ∈ Λ such that q ∈ ∂K_{i}, ⟨ν(q), v⟩ > 0, where ν(q) is the outward unit normal to ∂K at q, and the first reflection point of the billiard trajectory {pr_{1}(ϕ_{t}(x)) : t > 0} belongs to ∂K_{j}. Define ω : ∪_{i=1}^{k}R_{i} ⟶ M by ω(q, v) = ϕ_{−t}(q, v), where t = t(q, v) > 0 is the minimal number with pr_{1}(ϕ_{−t}(q, v)) ∈ ∂K. Using the construction of Bowen starting with appropriate cross-sections, one gets a Markov family R which satisfies the following extra condition:

(e) For every i = 1, …, k there exist s(i), t(i) ∈ {1, …, k_{0}} such that ω(R_{i}) ⊂ Π_{s(i)t(i)} and for any x ∈ R_{i} the trajectory ϕ_{[0,τ(x))}(x) has at most one reflection point.

It follows from a result of C. Robinson [Ro] that there exists an open neighbourhood V of Λ_{ϵ} in S^{∗}(Ω) and C^{1} transverse foliations F^{u} and F^{s} on V such that W_{ϵ}^{u}(x) ∩ V = F^{u}(x) and W_{ϵ}^{s}(x) ∩ V = F^{s}(x) for any x ∈ Λ ∩ V. Fix a neighbourhood V and C^{1} foliations F^{u} and F^{s} with these properties.

Choosing χ > 0 sufficiently small, we may assume that our Markov family R satisfies the following additional condition:

(f) For each i the cross-section D_{i} (and therefore the rectangle R_{i}) is contained in V.

¿From now on we will assume that R = {R_{i}}_{i=1}^{k} is a fixed Markov family for ϕ_{t} satisfying the extra conditions (e) and (f). Then

U = ⋃_{i=1}^{k} U_{i} ,

is a finite disjoint union of compact curves in V.

Using the foliations F^{u} and F^{s} in V, assuming again that V is sufficiently small, we can extend the product [x, y] over the whole U_{i} × S_{i} for any i as follows. Given x ∈ U_{i} ⊂ W_{ϵ}^{u}(z_{i}) = F^{u}(z_{i}) ∩ V and y ∈ S_{i} ⊂ W_{ϵ}^{s}(z_{i}) = F^{s}(x_{i}) ∩ V, the sets F^{s}(x) and ϕ_{[−ϵ,ϵ]}(F^{u}(y)) intersect at exactly one point [x, y]. The image of the C^{1} map U_{i} × S_{i} ∋ (x, y) ↦ [x, y] is a 2-dimensional sumanifold of S_{ϵ}^{∗}(Ω) which will be denoted by R̃_{i} = [U_{i}, S_{i}]. Changing appropriately the cross-section D_{i}, we may assume^{4} that R̃_{i} ⊂ D_{i} for each i. The projection p : R̃ = ∪_{i=1}^{k}R̃_{i} ⟶ U along the leaves of F^{s} is C^{1}. Then the Poincaré map P : R̃ ⟶ ∪_{i=1}^{k}D_{i} and the corresponding root function τ : R̃ ⟶ [0, ∞) are well-defined and C^{1}. Thus, σ can be extended to a C^{1} map σ : U ⟶ U by the same formula: σ = p ∘ P. In the same way one observes that Δ(x, y) is well-defined and C^{1} for (x, y) ∈ ∪_{i=1}^{k}R̃_{i} × R̃_{i}.

Convention. Given a subset A of U and an integer n > 0 we will denote by σ^{n}(A) the set of all well-defined σ^{n}(x) with x ∈ A. That is, σ^{n}(A) = σ^{n}(A ∩ σ^{−n}(U)).

A conjugacy between Λ∩R and the Bernoulli shift on a symbolic space Σ_{A} is now defined in the usual way. Let A = (A_{ij})_{i,j=1}^{k} be the matrix given by A_{ij} = 1 if P(R_{i}) ∩ R_{j} ≠ ∅ and A_{ij} = 0 otherwise. Then

Σ_{A} = {(i_{j})_{j=−∞}^{∞} : 1 ≤ i_{j} ≤ k, A_{i_{j} i_{j+1}} = 1 for all j },

4This is just for convenience; practically one does not need it.

and the Bernoulli shift σ̃ : Σ_{A} ⟶ Σ_{A} is given by σ̃((i_{j})) = ((i′_{j})), where i′_{j} = i_{j+1} for all j. Define Φ : R ⟶ Σ_{A} by Φ(x) = (i_{j})_{j=−∞}^{∞}, where P^{j}(x) ∈ R_{i_{j}} for all j ∈ Z. Notice that P : R ⟶ R is invertible, so P^{j} is well-defined for all integers j. The map Φ is a homeomorphism when Σ_{A} is considered with the product topology, and σ̃ ∘Φ = Φ∘P. The projection r of τ on Σ_{A} is given by r ∘ Φ = τ.

The subset Λ ∩ U of R can be naturally identified with

Σ_{A}^{+} = { (i_{j})_{j=0}^{∞} : 1 ≤ i_{j} ≤ k , A_{i_{j} i_{j+1}} = 1 for all j ≥ 0 }.

Namely, if π : Σ_{A} ⟶ Σ_{A}^{+} is the natural projection, then Φ^{+} = π ∘ Φ_{|Λ∩U} : Λ ∩ U ⟶ Σ_{A}^{+} is a bijection with σ̃ ∘ Φ^{+} = Φ^{+} ∘ σ, where σ̃ denotes the corresponding Bernoulli shift on Σ_{A}^{+}.

The suspended flow σ̃^{r} over Σ_{A}^{r} = {(ξ, t) : ξ ∈ Σ_{A}, 0 ≤ t ≤ r(ξ) } is defined by σ̃_{s}^{r}(ξ, t) = (ξ, t + s), where we use the identification (ξ, r(ξ)) = (σ(ξ), 0) in Σ_{A} × IR. Then Σ_{A}^{r,+} = {(ξ, t) ∈ Σ_{A}^{r} : ξ ∈ Σ_{A}^{+}} is a closed subset of Σ_{A}^{r} which is invariant under the semiflow σ̃_{t}^{r}, t ≥ 0 (cf. [PP2] for details). A conjugacy between σ̃_{t}^{r} and ϕ_{t} on Λ is defined by Φ^{r}(ϕ_{s}(x)) = (Φ(x), s) for x ∈ R and 0 ≤ s ≤ τ(x).

Given θ ∈ (0, 1), consider the metric d_{θ} on Σ_{A} given by d_{θ}(ξ, η) = 0 if ξ = η and d_{θ}(ξ, η) = θ^{n} if ξ_{i} = η_{i} for |i| ≤ n and n is maximal with this property. In a similar way one defines a metric d_{θ} on Σ_{A}^{+}. Using it, we get a metric on Σ_{A}^{r,+} by setting d_{θ}^{r}((ξ, t), (η, s)) = d_{θ}(ξ, η) + |t − s|^{θ}. The spaces of Lipschitz functions on Σ_{A}^{+} and Σ_{A}^{r,+} with respect to the metrics just defined will be denoted by F_{θ}(Σ_{A}^{+}) and F_{θ}(Σ_{A}^{r,+}), respectively.

Let F ∈ F_{α}(Λ) for some α > 0. Denote by F̃ the function on Σ_{A}^{r} such that F̃ ∘ Φ^{r} = Φ^{r} ∘ F. There exists θ = θ(α) ∈ (0, 1) such that F̃ ∈ F_{θ}(Σ_{A}^{r}). Let μ̃ be the Gibbs measure related to F̃ with respect to the suspended flow σ̃^{r}, and let P = Pr_{}_{σ̃^{r}}(F̃) be the topological pressure of F̃ with respect to σ̃^{r}. A function f̃ ∈ F_{θ}(Σ_{A}^{+}) related to F̃ is defined by

f̃(ξ) = ∫_{0}^{r(ξ)} F̃(ξ, s) ds .

Let ν̃ be the Gibbs measure on Σ_{A} determined by the function f̃− P r. Then (cf. [PP2])

dμ̃(ξ, s) = \frac{1}{ν(r)}dν̃(ξ)ds , where ν(g̃) = ∫_{Σ_{A}} g̃(ξ) dν̃(ξ) .

Moreover, we have Prσ̃(f̃− P r) = 0. The Gibbs measures μ̃ and ν̃ give rise to measures μ and ν on R and Λ ∩ U, respectively, via the conjugacies Φ^{r} and Φ. If f is the function on

Λ ∩ U such that f̃∘ Φ = Φ ∘ f, then f(x) = ∫_{0}^{τ(x)} F (ϕ_{s}(x)) ds, so f ∈ F_{α}(Λ ∩ U). Thus, according to Kirszbraun’s theorem (cf. Ch. 3 in [WW]), f has an extension to a function f ∈ F_{α}(U) with the same Hölder constant ∥f∥_{α} = ∥f_{|Λ∩U}∥_{α}. The measures μ and ν are called the Gibbs measures related to F and f − P τ, respectively. It follows from above that Pr_{ϕ_{t}}(F) = P and Pr_{σ}(f − P τ) = 0.

Let C(U) be the space of bounded continuous functions on U. For g ∈ C(U) denote ∥g∥ = ∥g∥_{0} = sup_{x∈U} |g(x)|. Given g, the Ruelle operator L_{g} : C(U) ⟶ C(U) is defined in the usual way:

(L_{g}h)(u) = ∑_{σ(v)=u} e^{g(v)}h(v).

If g ∈ C^{1}(U), then clearly L_{g} preserves the space C^{1}(U). Here C^{1}(U) denotes the space of differentiable function with bounded derivatives on U. For g ∈ C^{1}(U) we will denote by ∥dg(u)∥ the norm of the linear map dg(u) : T_{u}U ⟶ IR with respect to the Euclidean norm on T_{u}U ⊂ IR^{4}.

Given f ∈ F_{α}(U) and b ∈ IR with |b| sufficiently large, following [D], denote by f^{(b)} the average of f over subintervals of U of length 1/\sqrt{|b|}. To be more precise, one has to first fix an arbitrary extension f ∈ F_{α}(S^{∗}(Ω)) (with the same Hölder constant), then extend each connected component U_{i} of U appropriately (so every point of U_{i} is far enough from the ends of the extension) and then take the averages in question. Then f^{(b)} ∈ C^{1}(U), and as mentioned in [D], it has the following properties: (a) ∥f − f^{(b)}∥_{0} ≤ ∥f∥_{α}/|b|^{α/2}; (b) ∥df^{(b)}∥ ≤ const ∥f∥_{α} |b|^{(1−α)/2} ≤ const ∥f∥_{α} \sqrt{|b|}; (c) ∥f − f^{(b)}∥_{β} → 0 as |b| → ∞ for any β < α. In the special case f ∈ C^{1}(U) we set f^{(b)} = f for all b.

It follows from Theorem 3.6 in [R2] that if g ∈ C^{1}(U), then L_{g} : C^{1}(U) ⟶ C^{1}(U) has an eigenvalue λ_{g} equal to the spectral radius of L_{g} and a corresponding (positive) eigenfunction. Given real numbers a and b (with |a| + \frac{1}{|b|} small), denote by λ_{ab} the largest eigenvalue of L_{f^{(b)}−(P +a)τ} and by h_{ab} the corresponding (positive) eigenfunction such that sup_{u∈U} h_{ab}(u) = 1. The above remark implies that h_{ab} ∈ C^{1}(U). As in [D], one shows that for |a| + \frac{1}{|b|} sufficiently small we have ∥d_{u} ln h_{ab}(u)∥ ≤ const \sqrt{|b|} for all u ∈ U. Since Pr(f − P τ) = 0, it follows from the main properties of pressure (cf. e.g. Ch. 3 in [PP2]) and the properties of the approximation f^{(b)} mentioned above that | Pr(f^{(b)} − (P + a)τ)| ≤ const (|a| + \frac{1}{|b|^{α/2}} ), where const > 0 is a constant depeding on f. Thus, for small |a| and large |b|, the maximal eigenvalues λ_{ab} and the eigenfunctions h_{ab} are uniformly close to 1. In particular, for small |a| and large |b| we have h_{ab}(u) ≥ \frac{1}{2} for all u ∈ U.

Given real numbers a and b, as in [D], consider the function

f^{(ab)}(u) = f^{(b)}(u) − (P + a)τ(u) + ln h_{ab}(u) − ln h_{ab}(σ(u)) − ln λ_{ab}

and the operators

L_{ab} = L_{f^{(ab)}} : C(U) ⟶ C(U) , M_{ab} = L_{f^{(ab)}} : C(U) ⟶ C(U).

One checks that M_{ab} 1 = 1.

For a connected curve I in S^{∗}(Ω) we will denote by |I| the arc length of I with respect to the standard metric in S^{∗}(Ω) ⊂ IR^{2} × S^{1}. Given x ∈ U_{i} for some i and r > 0 we will denote by I(x, r) the set of all y ∈ U_{i} with dist_{U}(x, y) ≤ r, where dist_{U}(x, y) is the arc length of the connected subcurve of U_{i} with end points x and y.

It is well known (see also Sect. 5 below) that σ^{n} is an expanding map (on its domain σ^{−n}(U)) for sufficiently large integers n. Since σ is C^{1}, shrinking slightly R if necessary (and therefore U), it is uniformly Lipschitz on U. Thus, there exist constants c_{0} > 0 and γ_{1} > γ > 1 such that

(2.1) c_{0}γ^{n} dist_{U}(u_{1}, u_{2}) ≤ dist_{U}(σ^{n}(u_{1}), σ^{n}(u_{2})) ≤ \frac{γ_{1}^{n}}{c_{0}} dist(u_{1}, u_{2})

whenever u_{1} and u_{2} belong to the same connected component of U and σ^{n}(u_{i}), i = 1, 2, is well-defined. Set C_{0} = \frac{1}{c_{0}(γ−1)} . ¿From now on we will assume that c_{0}, C_{0}, γ and γ_{1} are fixed constants with the above properties.

Lemma 2.1. (Lemma 3 in [D]) Let h ∈ C^{1}(U) and let v(u) be a C^{1} map defined on some open subset V of U and such that σ^{n}(v(u)) = u for all u ∈ V. Then

∥dv(u)∥ ≤ \frac{1}{c_{0} γ^{n}} and ∥d_{u}h_{n}(v(u))∥ ≤ C_{0} sup_{v∈U} ∥dh(v)∥ .

If h ∈ F_{α}(U) for some α ∈ (0, 1], then ∥h_{n} ∘ v∥_{α} ≤ C_{0}∥h∥_{α}.

Set τ̃ = sup_{u∈U} τ(u). Given a sufficiently small number a′_{0} > 0, there exists T = T (a′_{0}) > 0 such that

T ≥ sup_{u∈U} |f^{(ab)}(u)| , T |b| ≥ sup_{u∈U} ∥df^{(ab)}(u)∥ (2.2)

for all |a| ≤ a′_{0}, 1/|b| ≤ a′_{0}. Fix a′_{0} > 0 and T > 0 and with these properties.

Lemma 2.2. (Lemma 2 in [D]) There exists a constants A_{0} > 0 such that for all a ∈ IR with |a| ≤ a′_{0} and all positive integers n we have:

(a) |(L_{ab}^{n}h)(u)| ≤ (M_{ab}^{n}|h|)(u) for all u ∈ U and h ∈ C(U);

(b) ∥d_{u}(L_{ab}^{n}h)(u)∥ ≤ A_{0}[ |b|(M_{ab}^{n}|h|)(u) + \frac{1}{γ^{n}} (M_{ab}^{n}∥dh∥)(u) ] for all u ∈ U and h ∈

C^{1}(U);

(c) ∥d_{u}(M_{ab}^{n}h)(u)∥ ≤ A_{0}[ |b|(M_{ab}^{n}|h|)(u) + \frac{1}{γ^{n}} (M_{ab}^{n}∥dh∥)(u) ] for all u ∈ U and h ∈ C^{1}(U);

(d) ∥L_{ab}^{n}h∥_{β} ≤ A_{0}[ |b| ∥h∥_{0} + \frac{1}{γ^{nβ}} ∥h∥_{β} ] for all u ∈ U and h ∈ F_{β}(U), β > 0.

3 The main estimate

The central point in this paper is to prove the L_{ν}^{1}-contraction property of Dolgopyat’s normalized operator L_{ab} with respect to the norm ∥h∥_{1,b} defined by ∥h∥_{1,b} = ∥h∥_{0} + \frac{1}{|b|}∥dh∥_{0} for |b| ≥ 1. The main results stated in Sect. 1 are consequences of this estimate. We will also use the notation ∥h∥_{0,U∩Λ} = sup_{u∈U∩Λ} |h(u)| and ∥h∥_{1,b,U∩Λ} defined as above replacing h by h_{|U∩Λ}.

Theorem 3.1. There exist a positive integer N and constants ρ_{1} ∈ (0, 1) and a_{0} > 0 such that for any a, b ∈ IR with |a| ≤ a_{0} and |b| ≥ 1/a_{0} and every h ∈ C^{1}(U) with ∥h∥_{1,b} ≤ 1 we have

∫_{U} |L_{ab}^{Nn}h|^{2} dν ≤ ρ_{1}^{n}

for every positive integer n, where ν is the Gibbs measure determined by f − P τ on U.

Remark 3.2. Since ν is a probability measure on U, combining the above with the Cauchy-Schwartz inequality, gives ∫_{U} |L_{ab}^{Nn}h| dν ≤ ρ_{1}^{n/2} for ∥h∥_{1,b} ≤ 1 and n > 0. Hence

∫_{U} |L_{ab}^{Nn}h| dν ≤ ρ_{1}^{n/2}∥h∥_{1,b} for any positive integer n and any h ∈ C^{1}(U). On the other hand, it follows from Whitney’s extension theorem (cf. Ch. 1 in [Mal]) that there exists a constant D > 0 such that for every h ∈ C^{1}(U) there is h̃ ∈ C^{1}(U) with h̃_{|U∩Λ} = h_{|U∩Λ}, (dh̃)_{|U∩Λ} = (dh)_{|U∩Λ} and ∥h̃∥_{0} ≤ D sup_{u∈U∩Λ} |h(u)| and ∥dh̃∥_{0} ≤ D sup_{u∈U∩Λ} ∥dh(u)∥. Hence in the above statement the norm ∥.∥_{1,b} in C^{1}(U) can be replaced by the seminorm (for |b| ≥ 1)

∥h∥_{1,b,U∩Λ} = ∥h∥_{0,U∩Λ} + \frac{1}{|b|}∥dh∥_{0,U∩Λ} .

That is, we have ∫_{U} |L_{ab}^{Nn}h| dν ≤ Dρ_{1}^{n/2} for all n = 1, 2, … and h ∈ C^{1}(U) with ∥h∥_{1,b,U∩Λ} ≤ 1. In the case when h ∈ F_{β}(U ∩ Λ) for some β > 0 we have a similar inequality involving the norm

∥h∥_{β,b,U∩Λ} = ∥h∥_{0,U∩Λ} + \frac{1}{|b|}∥h∥_{β,U∩Λ} .

Indeed, according to Kirszbraun’s theorem (cf. Ch. 3 in [WW]), such h can always be extended to a function h ∈ F_{β}(U) with the same Hölder constant ∥h∥_{β} = ∥h∥_{β,U∩Λ} = ∥h_{|U∩Λ}∥_{β}. Approximating h by functions h^{(p)} (p → ∞) from C^{1}(U) (cf. Sect. 2), one gets

∫_{U} |L_{ab}^{Nn}h| dν ≤ Dρ_{1}^{n/2} ∥h∥_{β,b,U∩Λ} (n = 1, 2, …, h ∈ F_{β}(U ∩ Λ)) . (3.1)

Theorem 1.1 follows immediately from part (a) of the following consequence of Theorem 3.1.

Corollary 3.3. (a) For every ϵ > 0 there exist constants ρ ∈ (0, 1), a_{0} ∈ (0, 1) and C > 0 such that for every integer n > 0, every β > 0 and every h ∈ F_{β}(U ∩ Λ), if a, b ∈ IR, are such that |a| ≤ a_{0}, 1/|b| ≤ a_{0}, then:

∥L_{ab}^{n}h∥_{β,b,U∩Λ} ≤ C ρ^{n} |b|^{ϵ} ∥h∥_{β,b,U∩Λ} and |L_{f^{(b)}−(P +a+ib)τ}^{n}h∥_{1,b,U∩Λ} ≤ C ρ^{n} |b|^{1/2} ∥h∥_{β,b,U∩Λ} .

When f ∈ C^{1}(U) and β ∈ (0, 1], under the same assumptions we have

|L_{f−(P +a+ib)τ}^{n}h∥_{β,b,U∩Λ} ≤ C ρ^{n} |b|^{ϵ} ∥h∥_{β,b,U∩Λ} .

(b) If β ≤ α, there exist constants ρ ∈ (0, 1), a_{0} ∈ (0, 1), ϵ′ > 0, B > 0 and C > 0 such that for h ∈ F_{β}(U ∩ Λ) and a, b ∈ IR with |a| ≤ a_{0} and 1/|b| ≤ a_{0} we have

∥L_{f−(P +a+ib)τ}^{2[B ln |b|]}h∥_{β,b,U∩Λ} ≤ \frac{C}{|b|^{ϵ′}} ∥h∥_{β,b,U∩Λ} .

The derivation of Corollary 3.3 from Theorem 3.1 essentially repeats the proofs of Corollaries 2 and 3 in [D] (cf. also Sect. 6 in [PoS]).

Proof of Corollary 3.3. (a) Let ρ_{1} ∈ (0, 1) and N be as in Theorem 3.1, and let ϵ > 0. We will assume that ρ_{1} ≥ \frac{1}{γ}, where γ > 1 is determined in Sect. 2. Again we will denote by const a global constant (independent of a, b, n). Let a, b ∈ IR be such that |a| ≤ a_{0} and |b|^{−α/2} ≤ a_{0}. As in Sect. 2, α > 0 denotes the Hölder exponent of f, i.e. we have f ∈ F_{α}(U). Without loss of generality we may assume that ϵ < 1. Set q = 1/ϵ and let p > 1 be such that \frac{1}{p} + \frac{1}{q} = 1.

Assume that h ∈ F_{β}(U ∩ Λ) for some β > 0 and ∥h∥_{β,b,U∩Λ} ≤ 1. According to Remark 3.2, we may assume that h ∈ F_{β}(U) with the same ∥h∥_{0} and ∥h∥_{β}. Since ν is the Gibbs measure for g = f − P τ on U and Pr(g) = 0, it folows from the Perron-Ruelle-Frobenius theorem that if we take ρ_{1} ∈ (0, 1) from Theorem 3.1 sufficiently close to 1 and const > 0 sufficiently large, then

|L_{g}^{m}h(u) − ∫_{U∩Λ} h dν| ≤ const ρ_{1}^{m} ∥h∥_{β} u ∈ U ∩ Λ , m = 1, 2, …. (3.2)

Lemma 2.2 and Hölder’s inequality imply

|L_{ab}^{2nN}h(u)| = |L_{ab}^{nN} (L_{ab}^{nN} ) (u)| ≤ M_{ab}^{nN} (|L_{ab}^{nN}h|) (u)
= L_{f^{(ab)}}^{nN} (|L_{ab}^{nN}h|) (u) = L_{g}^{nN} (e^{(f^{(ab)}−g)_{nN}} |L_{ab}^{nN}h|) (u)
(3.3)  ≤ [L_{g}^{nN} (e^{p(f^{(ab)}−g)_{nN}} ) (u)]^{1/p} [L_{g}^{nN} (|L_{ab}^{nN}h|^{q}) (u)]^{1/q} .

To estimate the first term, write L_{g}^{nN} (e^{p(f^{(ab)}−g)_{nN}} ) (u) = [L_{g+p(f^{(ab)}−g)}^{nN}1] (u). Let ĥ = ĥ_{a,b} be an eigenfunction of L_{g+p(f^{(ab)}−g)} with sup ĥ = 1. Since

| Pr(g + p(f^{(ab)} − g))| = | Pr([f − P τ] + p[f^{(b)} − f] − paτ) − p ln λ_{ab}| ≤ A_{1} (|a| + \frac{1}{|b|^{α/2}} )

for some constant A_{1} > 0 (depending on ϵ), it follows that for small |a| and large |b| we have ĥ(u) ≥ 1/2 for all u ∈ U ∩ Λ. Moreover, the Perron-Ruelle-Frobenius theorem implies

[L_{g+p(f^{(ab)}−g)}^{nN}1] (u) ≤ 2 [L_{g+p(f^{(ab)}−g)}^{nN}ĥ] (u) = 2e^{nN Pr(g+p(f^{(ab)}−g))}ĥ(u) ≤ 2e^{2A_{1}a_{0}nN}.

This estimate, q > 1, ∥h∥_{0} ≤ 1, (3.1) and (3.2) imply that for u ∈ U ∩ Λ we have

L_{g}^{nN} (|L_{ab}^{nN}h|^{q}) (u) ≤ L_{g}^{nN} (|L_{ab}^{nN}h|) (u) ≤ ∫_{U∩Λ} |L_{ab}^{nN}h| dν + const ρ_{1}^{nN}∥L_{ab}^{nN}h∥_{β}
≤ const ρ_{1}^{n/2} + const ρ_{1}^{nN}∥L_{ab}^{nN}h∥_{β} .

Assuming that 1/γ ≤ ρ_{1} and using Lemma 2.2(d), one gets ∥L_{ab}^{nN}h∥_{β,U∩Λ} ≤ A_{0}|b| ∥h∥_{β,b} ≤ const |b|. Thus, L_{g}^{nN} (|L_{ab}^{nN}h|^{q}) (u) ≤ const ρ_{1}^{n/2} |b| for all u ∈ U ∩ Λ. Combining this with (3.3), one gets

(3.4) ∥L_{ab}^{2nN}h∥_{0,U∩Λ} ≤ const e^{2A_{1}nNa_{0}/p}ρ_{1}^{n/2q} |b|^{1/q} ≤ const ρ_{2}^{n} |b|^{ϵ} ,

where ρ_{2} = e^{2A_{1}Na_{0}/p}ρ_{1}^{1/2q} < 1 provided a_{0} > 0 is sufficiently small.

Next, applying Lemma 2.2(d) again and using (3.4) with 2nN replaced by nN, we get

∥L_{ab}^{2nN}h∥_{β,U∩Λ} = ∥L_{ab}^{nN}(L_{ab}^{nN}h)∥_{β,U∩Λ} ≤ A_{0} (|b| ∥L_{ab}^{nN}h∥_{0,U∩Λ} + ρ_{1}^{nNβ}∥L_{ab}^{nN}h∥_{β,U∩Λ})
≤ const |b|^{1+ϵ} ρ_{2}^{n/2} + const ρ_{1}^{nNβ} [|b| ∥h∥_{0} + ρ_{1}^{nNβ}∥h∥_{β}]
≤ const |b|^{1+ϵ} ρ_{2}^{n/2} + const ρ_{1}^{nNβ} |b| ∥h∥_{1,b} ≤ const |b|^{1+ϵ}ρ_{3}^{n} ,

where ρ_{3} = max\{ρ_{2}^{1/2}, ρ_{1}^{β}\}. Thus, \frac{1}{|b|}∥L_{ab}^{2nN}h∥_{β,U∩Λ} ≤ const |b|^{ϵ} ρ_{3}^{n}. This and (3.4) yield

∥L_{ab}^{2nN}h∥_{β,b,U∩Λ} ≤ const |b|^{ϵ} ρ_{3}^{n} (3.5)

for all positive integers n and all h ∈ F_{β}(U ∩ Λ) with ∥h∥_{β,b,U∩Λ} ≤ 1. Taking ρ = ρ_{3}, proves the first part of (a).

To prove the second inequality in (a), notice that

[L_{ab}^{n}(h/h_{ab})](u) = ∑_{σ^{n}v=u} e^{f_{n}^{(b)}(v)−(P +a+ib)τ_{n}(v)+(ln h_{ab}−ln(h_{ab}∘σ))_{n}(v)−n ln λ_{ab}} \frac{h(v)}{h_{ab}(v)}
= \frac{1}{λ_{a}^{n}h_{ab}(u)} ∑_{σ^{n}v=u} e^{f_{n}^{(b)}(v)−(P +a+ib)τ_{n}(v)} h(v) = \frac{1}{λ_{ab}^{n}h_{ab}(u)}[L_{f^{(b)}−(P +a+ib)τ}^{n}h](u) .

Let β ∈ (0, 1] and h ∈ F_{β}(U ∩ Λ) be such that ∥h∥_{β,b,U∩Λ} ≤ 1. We will again consider an extension h ∈ F_{β}(U) with the same norms. Let h̃ = h/h_{ab}. Since ∥d_{u}h_{ab}(u)∥ ≤ const \sqrt{b} as remarked in Sect. 2, we have ∥h_{a,b}∥_{β} ≤ const \sqrt{b} which implies ∥h̃∥_{β,b} ≤ const . Using

this, (3.5) with h replaced by h̃/const and the fact that λ_{ab} ≤ e^{const (|a|+ \frac{1}{|b|^{α/2}} )} (see above), it follows that ∥L_{f^{(b)}−(P +a+ib)τ}^{n}h∥_{β,b,U∩Λ} ≤ const |b|^{1/2} ρ_{4}^{n} , (3.6)

where ρ_{4} = ρ_{3} e^{const a_{0}} < 1, provided a_{0} > 0 is sufficiently small. So, for the second inequality in (a) we can take ρ = ρ_{4}.

Finally, notice that when f ∈ C^{1}(U) we have (by definition) f^{(b)} = f for all b, so both λ_{ab} and h_{ab} do not depend on b. This implies ∥d_{u}h_{ab}(u)∥ ≤ const , ∥h_{a,b}∥_{β} ≤ const and λ_{ab} ≤ const . Now the above argument and (3.5) give ∥L_{f−(P +a+ib)τ}^{n}h∥_{β,b,U∩Λ} ≤ const |b|^{ϵ} ρ_{4}^{n} for all h ∈ F_{β}(U ∩ Λ) with ∥h∥_{β,b,U∩Λ} ≤ 1.

(b) This follows from (3.6) using the argument from Appendix 2 in [D]. We omit the details.

Given A > 0, define

K_{A} = K_{A}(U) = {h ∈ C^{1}(V) : h > 0, ∥d_{u}(ln h(u))∥_{V} ≤ A for all u ∈ U} .

Notice that if h ∈ K_{A}, then | ln h(u) − ln h(v)| ≤ A dist(u, v), which implies

(3.7) e^{−A dist(u,v)} ≤ \frac{h(u)}{h(v)} ≤ e^{A dist(u,v)} , u, v ∈ U.

The following lemma is the analogue of Lemma 10′′ in [D] ^{5}.

Lemma 3.4. There exist a positive integer N and constants ρ ∈ (0, 1), a_{0} > 0 and E > 0 such that for every a, b ∈ IR with |a| ≤ a_{0}, 1/|b| ≤ a_{0}, there exist an operator N = N (a, b) : C^{1}(U) ⟶ C^{1}(U) with the following properties:

(a) The operator N preserves the cone K_{E|b|} ; (b) For all H ∈ K_{E|b|} we have

5Lemma 10” in [D] is slightly more general.

∫_{U} |N H|^{2} dν ≤ ρ ∫_{U} H^{2} dν .

(c) If h, H ∈ C^{1}(U) are such that |h(u)| ≤ H(u) and ∥dh(u)∥ ≤ E|b|H(u) for all u ∈ U and H ∈ K_{E|b|}, then |L_{ab}^{N}h(u)| ≤ (N H)(u) and ∥d(L_{ab}^{N}h)(u)∥ ≤ E|b|(N H)(u) for all u ∈ U.

Theorem 3.1 is an easy consequence of Lemma 3.4. The proof of the fact is exactly the same as in [D] and we omit it.

Sections 4–8 are devoted to the proof of Lemma 3.4.

4 Distortion bounds

In order to apply Dolgopyat’s construction of the operators N (a, b) (cf. Lemma 3.4), we need some information roughly speaking about the distribution of points of U′ ∩ Λ in U′ for each (maximal in a sense) local unstable manifold U′ of the flow ϕ_{t}. More precisely, given such U′ with U′ ∩ Λ ≠ ∅, it is clear that U′ ∩ Λ is a Cantor set which is obtained by recursively applying the following procedure: given an nth generation interval (a connected subcurve of U′), divide it (in a special way) into 2k_{0} − 1 subintervals and throw away the odd numbered intervals. One of the aims of this section is to prove that the ratio of lengths of any two of these 2k_{0} − 1 subintervals is bounded by positive constants from below and above. This shows that the natural homeomorphism S between U′ ∩ Λ and the corresponding (classical in a sense) Cantor subset C(k_{0}) of [0, 1] has some ”Lipschitz-like” property. It should be mentioned however that in general S is not a Lipschitz equivalence^{6}.

Denote by M the set of those (q, v) ∈ S_{∂K}^{∗}(Ω) such that ⟨v, ν(q)⟩ > 0 whenever q ∈ ∂Ω = ∂K. A natural cross-section map for ϕ_{t} is the billiard ball map B defined on the subset M_{0} = Λ ∩ M of M. Clearly M_{0} is the maximal invariant set for B.

First, we describe a standard symbolic representation of the billiard ball map B on M_{0} which has some special features and in some cases is more convenient than a symbolic representation coming from an arbitrary Markov family.

Let A be the k_{0} × k_{0} matrix with entries A(i, j) = 1 if i ≠ j and A(i, i) = 0 for all i = 1, 2, …, k_{0}. Denote by Σ_{A} the space of all infinite in both directions sequences

ξ = (…, ξ_{−i}, ξ_{−i+1}, …, ξ_{−1}; ξ_{0}, ξ_{1}, …, ξ_{i−1}, ξ_{i}, …)

such that ξ_{i} ∈ {1, 2, …, k_{0}} and A(ξ_{i}, ξ_{i+1}) = 1 for all i (the latter simply means ξ_{i} ≠ ξ_{i+1}). Notice that ; separates the “negative” and “non-negative” parts of ξ. In a similar way one defines Σ_{A}^{+}, the space of one-sided sequences ξ = (ξ_{0}, ξ_{1}, …) with the above properties.

The natural symbolic representation map Q : M_{0} ⟶ Σ_{A} is defined by Q(x) = ξ, where ξ = (ξ_{n})_{n} is such that pr_{1}(B^{j}(x)) ∈ ∂K_{ξ_{j}} for all j ∈ Z. It is well-known that Q is a homeomorphism and Q ∘ B = σ_{A} ∘ Q, where σ_{A} : Σ_{A} ⟶ Σ_{A} is the shift map: (σ_{A}(ξ))_{n} = ξ_{n+1}.

6Indeed, otherwise the Hausdorff dimension dim_{H}(U′ ∩ Λ) would have been the same for all U′, and so dim_{H}(Λ) would be a positive constant depending on the number of connected components of K only but not on K itself. However, this is not the case; one can see that preserving the shape of the components of K and letting the distance between them tend to ∞ (in an appropriate fashion) makes dim_{H}(Λ) tend to 0. This is established rigorously in [K], where one can also find some estimates of dim_{H}(Λ) from below and above.

Given i ≠ j, (1 ≤ i, j ≤ k_{0}), denote by Π_{i,j} the set of those x = (q, v) ∈ M such that q ∈ ∂K_{i}, B(x) = (p, w) is defined and p ∈ ∂K_{j}. Following an old idea in symbolic dynamics (apparently due to Smale) the sets Π_{i,j} can be regarded as parallelograms of a Markov partition. Notice that Π̃_{i,j} = Q(M_{0} ∩ Π_{i,j}) = {ξ ∈ Σ_{A} : ξ_{0} = i, ξ_{1} = j}. Next, let V_{i,j} be the set of those (q, v) ∈ M such that B^{m}(q, v) = (q_{m}, v_{m}) is defined for all m ≤ 1, q_{0} = q ∈ ∂K_{i}, q_{m} ∈ ∂K_{i} for even m ≤ 0 and q_{m} ∈ ∂K_{j} for odd m ≤ 1. Clearly V_{i,j} is the local unstable set of the periodic point x = (p, w) of B with period 2 which corresponds to the shortest segment connecting K_{i} and K_{j}. Notice that

Ṽ_{i,j} = Q(M_{0} ∩ V_{i,j}) = {ξ = (…, i, j, i, j; i, j, ξ_{2}, ξ_{3}, …) ∈ Σ_{A}}.

Let π : Σ_{A} ⟶ Σ_{A}^{+} be the natural projection. We will denote by σ_{A} the Bernoulli shift on Σ_{A}^{+} and will identify Ṽ_{i,j} with its image π(Ṽ_{i,j}) in Σ_{A}^{+}. By diam(W) we denote the diameter of the subset W of S^{∗}(Ω) with respect to the metric ρ (cf. Sect. 2).

Given a finite string η = (η_{0}, η_{1}, …, η_{n}) of integers η_{i} = 1, 2, …, k_{0}, we will say that η is permissible if η_{i} ≠ η_{i+1} for i = 0, 1, …, n − 1. For such η one defines a cylinder

C̃[η] = {ξ ∈ Σ_{A}^{+} : ξ_{i} = η_{i} , i = 0, 1, …, n } ⊂ Σ_{A}^{+} .

Fix for a moment integers i_{0} ≠ j_{0} with 1 ≤ i_{0}, j_{0} ≤ k_{0}. It can be easily seen using a simple geometrical argument that the set Ṽ_{i_{0},j_{0}} ⊂ Σ_{A}^{+} is a cylinder, namely Ṽ_{i_{0},j_{0}} = C̃[(i_{0}, j_{0})]. Given a permissible string η = (η_{1}, …, η_{n}) with η_{1} ≠ j_{0}, denote

C_{i_{0},j_{0}}[η] = {x ∈ V_{i_{0},j_{0}} : Q(x) ∈ C̃[(i_{0}, j_{0}, η_{1}, …, η_{n})] },

and call this set a cylinder of length n in V_{i_{0},j_{0}} (later on we will call these sets big cylinders). One can easily see that C_{i_{0},j_{0}}[η] is a connected closed (i.e. containing its end points) curve in V_{i_{0},j_{0}}, and it is well-known that there exist a global constant μ ∈ (0, 1) (i.e. independent of i_{0}, j_{0}, n and η) such that

diam(C_{i_{0},j_{0}}[η]) ≤ const μ^{n} ,

for any η of length n, where const denotes another global positive constant (cf. [Si2], [BSC] or [I1], [Sj]). For any r = 1, …, k_{0} with r ≠ η_{n} the cylinder C_{i_{0},j_{0}}[(η, r)] of length n + 1 is a closed subcurve of C_{i_{0},j_{0}}[η]. Consider an arbitrary orientation of the curve C_{i_{0},j_{0}}[η]. Then, using again a simple geometrical argument, one can find a permutation {r_{1}, …, r_{k_{0}−1}} of {1, …, k_{0}}\{η_{n}} such that the curves C_{i_{0},j_{0}}[(η, r_{1})], C_{i_{0},j_{0}}[(η, r_{2})], …, C_{i_{0},j_{0}}[(η, r_{k_{0}−1})] follow successively one after another with respect to the fixed orientation of C_{i_{0},j_{0}}[η]. Denote by D_{1}, D_{2}, …, D_{k_{0}} the closed intervals in C_{i_{0},j_{0}}[η] that supplement the intervals C_{i_{0},j_{0}}[(η, r_{s})], i.e. D_{1}, C_{i_{0},j_{0}}[(η, r_{1})], D_{2}, C_{i_{0},j_{0}}[(η, r_{2})], …, D_{k_{0}−1}, C_{i_{0},j_{0}}[(η, r_{k_{0}−1})], D_{k_{0}} (4.1)

is an ordered sequence of intervals in C_{i_{0},j_{0}}[η] with disjoint interiors whose union equals C_{i_{0},j_{0}}[η]. So, the sequence (4.1) consists of ”gaps”, i.e. intervals without common points with the Cantor set M_{0}, and cylinders of length n + 1 contained in C_{i_{0},j_{0}}[η].

Most likely the following lemma can be derived from [BSC]. Here we give an elementary proof of it using Sinai’s estimates ([Si1], [Si2]) of the curvature of a convex wave front propagating in a dispersing billiard.

Lemma 4.1. There exists a global constant λ ∈ (0, 1) such that for any i_{0} ≠ j_{0}, n and η as above we have

λ |C_{i_{0},j_{0}}[η]| ≤ |C_{i_{0},j_{0}}[(η; r_{i})]| , λ |C_{i_{0},j_{0}}[η]| ≤ |D_{j}| (4.2)

for all i = 1, …, k_{0} − 1 and j = 1, …, k_{0}.

Proof. Take a small ϵ > 0 and consider the curve X = {q + ϵv : (q, v) ∈ C_{i_{0},j_{0}}[η] }. Then (cf. [Si1],[Si2]) X is a strictly convex smooth curve in the exterior of K in IR^{2} and for any x = q + ϵv ∈ X with (q, v) ∈ C_{i_{0},j_{0}}[η], the vector ν(x) = v coincides with the ”exterior” unit normal to X at x. Moreover, the map f : C_{i_{0},j_{0}}[η] ⟶ X, f(q, v) = q + ϵv, is a diffeomorphism. Denote X_{i} = f(C_{i_{0},j_{0}}[(η, r_{i})]), Y_{j} = f(D_{j}). Then Y_{1}, X_{1}, Y_{2}, X_{2}, …, Y_{k_{0}−1}, X_{k_{0}−1}, Y_{k_{0}} is an ordered family of closed curves in X (with respect to the orientation on X induced by C_{i_{0},j_{0}}[η]) with disjoint interiors and whose union equals X. Since f is smooth and X is smooth and convex, it follows that for any connected curve I on X we have const |f^{−1}(I)| ≤ |I| ≤ const |f^{−1}(I)|, where as before const denotes a global positive constant (which may be different in different experessions).

Fix an arbitrary i′ = 1, …, k_{0} − 1 and denote Z = X_{i′}, Y = Y_{i′}. We will prove that

(4.3) const ≤ |Z|/|Y | ≤ const .

In the same way one shows that const ≤ |X_{i′}|/|Y_{i′−1}| ≤ const . Applying these two inequalities finitely many (≤ k_{0}) times, one gets (4.2).

Denote the end points of the segments Z and Y by q(0), q(1) and q(1), q(2), respectively (then q(0), q(1), q(2) are ordered according to the orientation of X). According to the definition of Z, for every point q ∈ Z, the forward billiard trajectory generated by (q, ν(q)) makes at least n + 1 reflections and the jth reflection point belongs to ∂K_{η_{j}} for j ≤ n, while the (n + 1)st reflection point belongs to ∂K_{r_{i′}}. When q runs over Z the (n + 1)st reflection point of the corresponding trajectory describes a connected part Z′ of ∂K_{r_{i′}} with end points p(0) and p(1), where the trajectories of (q(0), ν(q(0))) and (q(1), ν(q(1))), respectively, have tangencies. Let p(s), 0 ≤ s ≤ 1, be the smooth parametrization of Z′ such that ∥ṗ(s)∥ = ℓ_{1} = |Z′| for all s ∈ [0, 1]. This induces a parametrization q(s), 0 ≤ s ≤ 1, of Z such that the (n + 1)st reflection point of the trajectory determined by (q(s), ν(q(s))) is p(s). Clearly q(s) is continuous for all s and smooth for s ∈ (0, 1).

When q is in the interior of Y, the forward billiard trajectory generated by (q, ν(q)) has only m reflection points and again the jth reflection point belongs to ∂K_{η_{j}}. If i′ < k_{0} − 1, denote by p(2) the (n + 1)st reflection point of the trajectory generated by (q(2), ν(q(2))); this is in fact a point of tangency to ∂K_{r_{i′+1}}. In the case when the trajectory generated by (q(2), ν(q(2))) has only n common points with ∂K, denote by p(2) the intersection point of this trajectory with the boundary of the convex hull of K. Let p(s), 1 ≤ s ≤ 2, be the smooth parametrization of the straightline segment [p(1), p(2)] in IR^{2} with ∥ṗ(s)∥ = ∥p(2) − p(1)∥ = ℓ_{2} ≤ 2diam(K). Then we get a corresponding parametrization q(s), 1 ≤ s ≤ 2, of Y which is continuous for 1 ≤ s ≤ 2 and smooth for 1 < s < 2.

Given s ∈ [0, 2], denote by q_{j}(s) (1 ≤ j ≤ n) the jth reflection point of the forward billiard trajectory γ(s) generated by (q(s), ν(q(s))) and set q_{0}(s) = q(s) and q_{n+1}(s) = p(s). Let κ_{j}(s) > 0 be the curvature of ∂K at q_{j}(s) and let φ_{j}(s) be the angle between the outward normal to ∂K at q_{j}(s) and the reflected direction of γ(s) at q_{j}(s). Set

d_{j}(s) = ∥q_{j+1}(s) − q_{j}(s)∥ , 0 ≤ j ≤ n.

Denote by k_{0}(s) > 0 the curvature of the curve X at q(s), define by induction

k_{j+1}(s) = \frac{k_{j}(s)}{1 + d_{j}(s)k_{j}(s)} + 2\frac{κ_{j}(s)}{cos φ_{j}(s)} , 0 ≤ j ≤ n, (4.4)

and set

δ_{j}(s) = \frac{1}{1 + d_{j}(s)k_{j}(s)} , 0 ≤ j ≤ n. (4.5)

Lemma 4.2.([St2]) For all s ∈ (0, 1) ∪ (1, 2) and all j = 0, 1, …, n − 1, we have^{7}

const ∥ṗ(s)∥δ_{j+1}(s)δ_{j+2}(s) … δ_{n}(s) ≤ ∥q̇_{j}(s)∥ ≤ const ∥ṗ(s)∥δ_{j+1}(s)δ_{j+2}(s) … δ_{n}(s) .

As one would expect, these inequalities follow from the chain rule for the iterations of the billiard ball map and estimates about the derivative of a single iteration ([Si2], [BSC]). For completeness we sketch a proof of Lemma 4.2 in the Appendix.

Now we continue with the proof of Lemma 4.1. Since ℓ_{1} and ℓ_{2} are bounded from below and above by positive constants^{8}, it follows that for s ∈ (0, 1) ∪ (1, 2), ∥ṗ(s)∥ is bounded from below and above by constants, so Lemma 4.2 implies

C′δ_{1}(s) … δ_{n}(s) ≤ ∥q̇(s)∥ ≤ C′′δ_{1}(s) … δ_{n}(s) , s ∈ (0, 1) ∪ (1, 2), (4.6)

for some constants C′′ > C′ > 0. Since the curvature of ∂K is bounded from below and above (by positive constants) and the distances d_{j}(s) are also uniformly bounded from below and above, it follows easily that there exist constants μ′ and μ with 0 < μ′ < μ < 1 such that

μ′ ≤ δ_{j}(s) ≤ μ , 0 ≤ s ≤ 2 , 1 ≤ j ≤ n .

Remark. Using these estimates for δ_{j}(s) in Lemma 4.2, one gets const μ^{′n−j} ≤ ∥q̇_{j}(s)∥ ≤ const μ^{n−j}, which are the usual distortion bounds well-known in hyperbolic dynamics. With j = 0 they give const μ^{′n} ≤ |Z| ≤ const μ^{n} and the same estimate for |Y |. Combining these two, one only gets the rough estimates const (μ′/μ)^{n} ≤ |Z|/|Y | ≤ const (μ/μ′)^{n} , which are clearly very far from what we want in (4.3), where ”const ” does not depend on n. To prove (4.3) one has to use Lemma 4.2 (or some similar tool) in a more careful way.

7For j = 0 one actually gets an exact formula: ∥q̇_{0}(s)∥ = ∥ṗ(s)∥δ_{1}(s)δ_{2}(s) … δ_{n}(s).

8For ℓ_{2} this follows immediately from its definition. For ℓ_{1} one can use for example a simple geometric argument involving min_{i≠j} dist(∂K_{i}, ∂K_{j}), diam(K) and the radius of the largest circle inscribed in K_{r_{i′}}.

Fix for a moment s ∈ (0, 1) and t ∈ (1, 2). We need to estimate the ratio δ_{j}(s)/δ_{j}(t) from below and above. First, using roughly the estimates in Lemma 4.2, one gets ∥q_{j}(s)−q_{j}(t)∥ ≤ const μ^{n−j} for all j = 1, …, n. (Here we use the fact that Z, Y ⊂ C_{i_{0},j_{0}}[η], so for any s′ ∈ [0, 2] the trajectory γ(s) has n consecutive reflection at the same boundary components ∂K_{η_{1}}, …, ∂K_{η_{n}}.) This implies

|d_{j}(s)−d_{j}(t)| ≤ const μ^{n−j} , |κ_{j}(s)−κ_{j}(t)| ≤ const μ^{n−j} , | cos φ_{j}(s)−cos φ_{j}(t)| ≤ const μ^{n−j}

for all j = 1, …, n. By induction, |k_{j}(s) − k_{j}(t)| ≤ const μ^{n−j} which implies |δ_{j}(s)/δ_{j}(t) − 1| ≤ const μ^{n−j} for all j = 1, …, n. Changing the roles of s and t, one gets similiar etsimates, so

\frac{δ_{j}(s)}{δ_{j}(t)} ≤ 1 + cμ^{n−j} , \frac{δ_{j}(t)}{δ_{j}(s)} ≤ 1 + cμ^{n−j} , j = 0, 1, …, n (4.7)

for some constant c > 0. It follows from the Mean Value Theorem that there exists s ∈ (0, 1) with |Z| = ∫_{0}^{1} ∥q̇(u)∥ du = ∥q̇(s)∥. In the same way, there exists t ∈ (1, 2) with |Y | = ∥q̇(t)∥. Then combining (4.6) and (4.7), one derives

\frac{|Z|}{|Y |} = \frac{∥q̇(s)∥}{∥q̇(t)∥} ≤ \frac{C′′δ_{1}(s) … δ_{n}(s)}{C′δ_{1}(t) … δ_{n}(t)} ≤ \frac{C′′}{C′} ∏_{j=1}^{n}(1 + cμ^{n−j}).

Hence log(|Z|/|Y |) ≤ log(C′′/C′) + ∑_{j=1}^{n} cμ^{n−j} < log(C′′/C′) + \frac{c}{1−μ}, and therefore \frac{|Z|}{|Y |} ≤ \frac{C′′}{C′} e^{c/(1−μ)}. In the same way, changing the roles of s and t again, we get \frac{|Y |}{|Z|} ≤ \frac{C′′}{C′} e^{c/(1−μ)}. This proves (4.3) and thus completes the proof of the lemma.

Next, we deal with the Markov family R = ∪_{i=1}^{k}R_{i} and the set U = ∪_{i=1}^{k}U_{i} defined in Sect. 2. Recall that each rectangle R_{i} is contained in a cross-section D_{i} to the flow ϕ_{t}. We will also use the map ω : ∪_{i=1}^{k}R_{i} ⟶ M defined in Sect. 2. The restriction of ω to any U_{i} is a diffeomorphism between U_{i} and ω(U_{i}). Consequently, there exist constants c_{2} > c_{1} > 0 such that for any connected curve I contained in some U_{i} we have

c_{1}|ω(I)| ≤ |I| ≤ c_{2}|ω(I)|. (4.8)

We will assume that c_{1} ≤ 1. Fix an arbitrary λ > 0 with the property described in Lemma 4.1. Clearly λ ≤ 1/(2k_{0} − 1). Setting

λ′ = \frac{c_{1}}{c_{2}} λ , λ_{0} = c_{1}λ^{2} ,

we have λ_{0} ≤ λ^{2} ≤ 1/(2k_{0}−1)^{2}. If Θ is a connected subcurve of U_{i} such that ω(Θ) = C_{i_{0},j_{0}}[η] for some i_{0} ≠ j_{0} and η = (η_{1}, …, η_{n}), we will say that Θ is a cylinder of length n in U_{i} ⊂ U.

The first part of the following Corollary will be used in Sect. 5, while the second one is needed for Sect. 6.

Corollary 4.3. (a) If Θ is a cylinder of length s + 1 contained in some connected component of U and Θ′ is a cylinder of length n + 1 with Θ′ ⊂ Θ (so n ≥ s), then

|Θ′| ≤ (1 − 2k_{0}λ′)^{n−s}|Θ|.

(b) For any x ∈ U ∩ Λ and any r > 0 with r ≤ dist_{U}(x, ∂U) there exists an ordered family {Θ_{j}}_{j=1}^{2k_{0}−1} of connected closed subcurves of I(x, r) having disjoint interiors such that:

(i) ∪_{j=1}^{2k_{0}−1}Θ_{j} is connected and |Θ_{j}| ≥ λ_{0}r for all j = 1, …, 2k_{0} − 1 ;

(ii)For any j = 1, …, 2k_{0} −1, the interior of Θ_{j} has common points with Λ iff j is even.

Proof. (a) Let ω(Θ) = C_{i_{0},j_{0}}[η′], ω(Θ′) = C_{i_{0},j_{0}}[η], where η = (η_{1}, …, η_{n}) and η′ = (η_{1}, …, η_{s}) is a part of η. Let {Θ_{j}}_{j=1}^{2k_{0}−1} be the sequence of cylinders of length s + 1 and gaps contained in C_{i_{0},j_{0}}[η′]. It follows from Lemma 4.1 that |Θ_{j}| ≥ λ|C_{i_{0},j_{0}}[η′]|. Hence the length of each curve of the form Θ′_{j} = (ω_{|U_{i}})^{−1}(Θ_{j}) is ≥ c_{1}λ |C_{i_{0},j_{0}}[η′]| ≥ λ′|Θ|. There exists exactly one j′ with Θ′ ⊂ Θ′′ = Θ′_{j′} ⊂ Θ. Hence

|Θ′′| = |Θ| − ∑_{j≠j′} |Θ′_{j}| ≤ |Θ| − 2k_{0}λ′|Θ| = (1 − 2k_{0}λ′)|Θ|.

In the same way, one shows that if Θ′′′ is the only cylinder of length s + 2 contained in Θ and containing Θ′, then |Θ′′′| ≤ (1 − 2k_{0}λ′)^{2}|Θ|. Continuing in this way by induction, one gets the desired inequality.

(b) Let x ∈ U ∩ Λ and r > 0 be such that r ≤ dist_{U}(x, ∂U). Let n be the minimal non-negative integer such that ω(I(x, r)) contains a cylinder C_{i_{0},j_{0}}[η] of length n. (The special form of the curve I(x, r) and x ∈ Λ show that I(x, r) ∩ Λ contains a non-trivial open subset of U ∩Λ.) Fix η = (η_{1}, …, η_{n}) with this property; then we have |C_{i_{0},j_{0}}[η]| ≤ 2r. Here we consider the case n ≥ 1 in detail; the case n = 0 follows in the same way (in fact it is an immediate consequence of the above lemma). The minimality of n shows that the cylinder C′ = C_{i_{0},j_{0}}[η_{1}, …, η_{n−1}] of length n − 1 containing C_{i_{0},j_{0}}[η] (and therefore x) is not contained in Θ = I(x, r). (When n = 1 we have the trivial case C′ = V_{i_{0},j_{0}}.) This is only possible when |C′| > r. Combining the latter with Lemma 4.1 and (4.8), it follows that

λ r < λ|C′| ≤ |C_{i_{0},j_{0}}[η]|. (4.9)

Let Θ′_{1}, Θ′_{2}, …, Θ′_{2k_{0}−1} be the sequence of ”gaps” and cylinders of length n + 1 contained in C_{i_{0},j_{0}}[η]. By the definition of these curves, we have that their union is C_{i_{0},j_{0}}[η], so it is connected, and the interior of Θ′_{j} has common points with M_{0} iff j is even. Moreover Lemma 4.1 and (4.8) imply that for Θ_{j} = (ω_{|U_{i}})^{−1}(Θ′_{j}) we have |Θ_{j}| ≥ c_{1}λ|C_{i_{0},j_{0}}[η]|. This and (4.9) give |Θ_{j}| ≥ c_{1}λ^{2}r = λ_{0} r for all j.

5 The Gibbs measure ν on U

Our aim in this section is to prove that the Gibbs measure ν on U is a Federer measure, that is, it has the following property.

Proposition 5.1. For every m > 0 there exists a constant D_{m} > 0 such that for all x ∈ U ∩ Λ and all r > 0 we have ν(I(x, mr)) ≤ D_{m} ν(I(x, r)).

We will use the notation from Sect. 2 and 4.

Now we have two symbolic representations at our disposal – the first one is Φ : R ⟶ Σ_{A} given by the Markov family R (cf. Sect.2), while the second one is Q : M_{0} ⟶ Σ_{A} determined by the standard ”Markov partition” Π = ∪Π_{i,j} (cf. Sect. 4). Correspondingly, we get two types of cylinders on U = ∪_{i=1}^{k}U_{i}. The second type are cylinders (we will call them big cylinders) in U that correspond, via the projection ω : U ⟶ M, to cylinders of the form C_{i_{0},j_{0}}[η] in Π. To describe the first type, consider a finite string ı = (i_{0}, i_{1}, …, i_{p}) of integers i_{j} ∈ {1, …, k}. We will say that ı is A-permissible if A_{i_{j}i_{j+1}} = 1 for any j = 0, 1, …, p − 1. Then ı defines a cylinder

C̃[ı] = {(i′_{j})_{j=0}^{∞} : i′_{j} = i_{j}, j = 0, 1, …, p} ⊂ Σ_{A}^{+} .

The set C[ı] of those x ∈ U such that σ^{j}(x) exists for all j = 0, 1, …, p and σ^{j}(x) ∈ U_{i_{j}}, j = 0, 1, …, p, will be called a small cylinder of length p in U. Clearly, C[ı] is a connected subcurve of U_{i_{0}} and (Φ^{+})(C[ı]) = C̃[ı]. Recall that the projection ω(D_{q}) of each crosssection D_{q} is contained in some Π_{s(q),t(q)} (s(q), t(q) ∈ {1, …, k_{0}}). So, x ∈ C[ı] means in particular that the first backward reflection of the billiard trajectory determined by x occurs at ∂K_{s(i_{0})}, while the first forward one occurs at ∂K_{t(i_{0})}. Then σ(x) ∈ D_{i_{1}} and the fact that between x and P(x) there is at most one reflection at ∂K, show that either s(i_{1}) = s(i_{0}) and t(i_{0}) = t(i_{1}) or s(i_{1}) = t(i_{0}). Continuing in this way, we see that for each j < p, either the pairs (s(i_{j}), t(i_{j})) and (s(i_{j+1}), t(i_{j+1})) are the same or t(i_{j}) = s(i_{j+1}). Transform the sequence (s(i_{0}), t(i_{0})), (s(i_{1}), t(i_{1})), …, (s(i_{p}), t(i_{p})) as follows: first remove the repeating successive pairs and then in the new sequence remove the repeating successive numbers. As a result we get a sequence of the form I(ı) = (i′_{0}, j′_{0}; η_{1}, …, η_{n}), where i′_{0} = s(i_{0}), j′_{0} = t(i_{0}), n ≥ p/k and η = (η_{1}, …, η_{m}) are such that ω(C[ı]) ⊂ C_{i′_{0},j′_{0}}[η]. This gives a map I from the set of finite A-permissible sequences onto the set of finite A-permissible sequences. Since |C_{i′_{0},j′_{0}}[η]| ≤ const μ^{n} for some constant μ ∈ (0, 1) (cf. Sect. 4), it follows that |C[ı]| ≤ const μ_{1}^{p}, where μ_{1} = μ^{1/k} ∈ (0, 1). In particular, σ^{p} : U ⟶ U is expanding for all sufficiently large p.

The above argument also implies that given an A-permissible sequence ζ = (i′_{0}, j′_{0}; η_{1}, …, η_{n}), we have C_{i′_{0},j′_{0}}[η] = ∪_{I(ı)=ζ} ω(C[ı]), where the union is taken over all A-permissible finite sequences ı with I(ı) = ζ. Choosing minimal sequences ı with this property, we get a representation of the big cylinder C_{i′_{0},j′_{0}}[η] as a finite disjoint union of small cylinders:

Lemma 5.2. For every A-permissible sequence ζ = (i′_{0}, j′_{0}; η_{1}, …, η_{n}) there exists a finite set ı^{(1)}, …, ı^{(ℓ)} of finite A-permissible sequences such that I(ı^{(j)}) = ζ for all j = 1, …, ℓ,

C_{i′_{0},j′_{0}}[η] = ∪_{j=1}^{ℓ}ω(C[ı^{(j)}]),

and C[ı^{(j)}] ∩ C[ı^{(j′)}] = ∅ for j ≠ j′.

Recall that ν is the Gibbs measure on U determined by the C^{1} function g = f − P τ. Denote

g_{0} = max_{z∈R} |g(z)|.

According to the definition of Gibbs measures (cf. [Si3] or [PP2]), there exist constants α > 0, β > 0 and c ∈ IR such that

α ≤ \frac{ν(C[ı])}{e^{pc}e^{g_{p}(x)}} ≤ β (5.1)

for any small cylinder C[ı] of length p in U and any x ∈ C[ı].

The main tool needed to establish the Federer property of ν is the following.

Lemma 5.3. There exists a positive integer ℓ_{0} such that for any positive integer n, any big cylinder C = C_{i_{0},j_{0}}[η_{1}, …, η_{n}] and any two maximal small cylinders C[ȷ] = C[j_{0}, j_{1}, …, j_{p}] and C[ȷ′] = C[j′_{0}, j′_{1}, …, j′_{p′}] whose ω-images are contained in C and C[ȷ] ∩ Λ ≠ ∅, C[ȷ′] ∩ Λ ≠ ∅, we have |p − p′| ≤ kℓ_{0} and j_{i} = j′_{i} for i ≤ min{p, p′} − kℓ_{0}. Consequently, the number of different maximal small cylinders whose ω-images are contained in C and have common points with Λ is ≤ k^{2kℓ_{0}} (which is independent of n). Moreover there exists a constant A_{1} > 0 such that for any two cylinders C[ȷ] and C[ȷ′] as above and any y ∈ C[ȷ] ∩ Λ and y′ ∈ C[ȷ′] ∩ Λ we have |g_{p}(y) − g_{p′}(y′)| ≤ A_{1}. Thus, ν(C[ȷ′]) ≤ A_{2} ν(C[ȷ]) for some constant A_{2} > 0 independent of n, p and p′.

Proof. As mentioned in Sect. 4 (this actually follows from Lemma 4.2), there exist constants μ ∈ (0, 1) and E > 0 such that for any big cylinder C of length ℓ we have |C| < Eμ^{ℓ}. The choice of the Markov family R = {R_{i}} shows that for each i = 1, …, k, R_{i} is contained in the interior of the generalized rectangle R̃_{i} = [U_{i}, S_{i}] (which is in D_{i}; cf. Sect. 2), so dist(∂R̃_{i}, R_{i}) > 0. Since the restriction of the map ω : ∪_{i=1}^{k}D_{i} ⟶ S_{∂K}^{∗}(Ω) to each D_{i} is a C^{1} diffeomorphism, it follows that there exists ϵ_{0} > 0 such that whenever y, y′ ∈ S_{∂K}^{∗}(Ω) ∩ Λ = M_{0} are such that y ∈ ω(R_{i}) for some i and dist(y, y′) < ϵ_{0}, then y′ ∈ ω(R̃_{i}), and so y′ ∈ ω(R_{i}), too. Fix a positive integer ℓ_{0} such that Eμ^{ℓ_{0}} < ϵ_{0}.

Let ȷ = (j_{0}, j_{1}, …, j_{p}) and ȷ′ = (j′_{0}, j′_{1}, …, j′_{p′}) be A-premissible sequences of minimal length such that ω(C[ȷ]) ⊂ C and ω(C[ȷ′]) ⊂ C. We will prove that |p′ − p| ≤ kℓ_{0} and j_{i} = j′_{i} for all i = 0, 1, …, p − kℓ_{0}. We assume p ≥ kℓ_{0} or p′ ≥ kℓ_{0}, otherwise there is nothing to prove. Assume for example that p ≥ kℓ_{0}. Fix arbitrary x ∈ ω(C[ȷ] ∩ Λ) and x′ ∈ ω(C[ȷ′]∩Λ). Then x ∈ ω(U_{j_{0}})∩Λ and x′ ∈ ω(U_{j′_{0}})∩Λ. The assumptions about C[ȷ] and C[ȷ′] imply that s(j_{0}) = s(j′_{0}) = i_{0}, t(j_{0}) = t(j′_{0}) = j_{0}. Set x_{i} = B^{i}(x), x′_{i} = B^{i}(x′), i ≥ 1, where as before B denotes the billiard ball map on M (cf. Sect. 4). Since x, x′ ∈ C and C is a big cylinder of length n, we have dist(x, x′) < Eμ^{n} ≤ ϵ_{0} if n ≥ l_{0}. So, the choice of ϵ_{0} gives that for n ≥ ℓ_{0} we have x ∈ ω(R_{i}) for some i iff x′ ∈ ω(R_{i}). Thus, the billiard trajectory of x between x and x_{1} and that of x′ between x′ and x′_{1} intersect the same set of rectangles R_{i}. Hence for all i with i_{0} = s(j_{0}) = s(j_{1}) = … = s(j_{i}) we have j_{i} = j′_{i}. Proceeding by induction, assume that for some i < p − kℓ_{0} we have j_{i′} = j′_{i′} for i′ = 0, 1, …, i. Let s(j_{i+1}) = η_{q} for some q = 1, …, n. If s(j_{i}) = s(j_{i+1}), then j_{i+1} = j′_{i+1} follows as above from the fact that the billiard trajectories of x and x′, between x_{q} and x_{q+1} and x′_{q} and x′_{q+1}, respectively, will intersect the same set of rectangles. Assume s(j_{i}) = s(j′_{i}) = η_{q−1}. Since between two reflection points x_{r} and x_{r+1} the billiard trajectory of x meets at most k rectangles R_{j} (k is the total number of rectangles), it follows that p − i ≤ k(n − q). This and i < p − kℓ_{0} imply n − q > ℓ_{0}. Since x_{q} and x′_{q} belong to the same cylinder of length n − q, it follows that dist(x_{q}, x′_{q}) < Eμ^{n−q} < Eμ^{ℓ_{0}} ≤ ϵ_{0}. Hence the billiard trajectory of x between x_{q} and x_{q+1} and that of x′ between x′_{q} and x′_{q+1} have common points with the same set of rectangles R_{j}. In particular, j_{i+1} = j′_{i+1}. This proves that j_{i} = j′_{i} for all i ≤ p − kℓ_{0}. By symmetry, this must be also true for all i ≤ p′ − kℓ_{0}. Hence |p − p′| ≤ kℓ_{0}.

Having fixed C[ȷ] and therefore p, the latter implies that for p′ we have at most 2kℓ_{0} different possible values. For each i > max{p − kℓ_{0}, p′ − kℓ_{0}}, there are at most k different possible values for j′_{i}. Thus, there are no more than k^{2kℓ_{0}} different possible sequences ȷ′.

Let C[ȷ] and C[ȷ′] be as above and let y ∈ C[ȷ] ∩ Λ and y′ ∈ C[ȷ] ∩ Λ. It follows from the C^{1} assumption about f that g is Lipschitz. Moreover, the choice of μ and the fact that y, y′ ∈ C[j_{0}, j_{1}, …, j_{p−kℓ_{0}}], imply that dist_{U}(σ^{i}(y), σ^{i}(y′)) < Eμ_{1}^{p−kℓ_{0}−i} for all i = 0, 1, …, p − kℓ_{0}, where μ_{1} = μ^{1/k} and E > 0 is a constant (which may be different from the one chosen above). Thus,

|g_{p}(y) − g_{p′}(y′)| ≤ ∑_{i=0}^{p−kℓ_{0}−1} |g(σ^{i}(y)) − g(σ^{i}(y′))| + ∑_{i=p−kℓ_{0}}^{p−1} |g(σ^{i}(y))| + ∑_{i=p−kℓ_{0}}^{p′−1} |g(σ^{i}(y′))|

≤ E ∥g∥_{1} ∑_{i=0}^{p−kℓ_{0}−1} μ_{1}^{p−kℓ_{0}−i} + kℓ_{0}g_{0} + 2kℓ_{0}g_{0} ≤ A_{1} = \frac{E ∥g∥_{1}}{1 − μ_{1}} + 3kℓ_{0}g_{0} ,

where ∥g∥_{1} is the Lipschitz constant of g.

Finally, using (5.1) we get

\frac{ν(C[ȷ])}{ν(C[ȷ′])} ≤ \frac{βe^{pc}e^{g_{p}(y)}}{αe^{p′c}e^{g_{p′}(y′)}} = \frac{βe^{(p−p′)c}e^{g_{p}(y)−g_{p′}(y′)}}{α} ≤ \frac{βe^{kℓ_{0}|c|}e^{A_{1}}}{α} .

So, we can take A_{2} = \frac{βe^{kℓ_{0}|c|+A_{1}}}{α}.

Proof of Proposition 5.1. Let m ≥ 1, and let x ∈ U ∩ Λ and r > 0. Denote by U′ the connected component of U that contains x.

First, assume that mr < dist_{U′}(x, ∂U′). Let ω(x) ∈ V_{i_{0},j_{0}}. Denote by n′ + 1 the minimal length of a big cylinder in V_{i_{0},j_{0}} containing ω(x) and contained in ω(I(x, mr)), and let C_{i_{0},j_{0}}[η_{1}, …, η_{n′}, η_{n′+1}] be a cylinder with this property. Set C′ = C_{i_{0},j_{0}}[η], where η = (η_{1}, …, η_{n′}). The minimality of n′ + 1 and the fact that ω(x) ∈ C′ imply that C′ is not contained in ω(I(x, mr)) which is only possible when |C′| > mr/c_{2} (cf. (4.8)). In general however C′ may not contain ω(I(x, mr)), so we still do not have an estimate from above for ν(I(x, mr)). To get such an estimate, notice that ω(I(x, mr)) can have common points with at most one more big cylinder of length n′ (if it has common points with three big cylinders of length n′, it must contain one of them). Assume that there exists a big cylinder Ĉ ≠ C′ of length n′ which has common points with ω(I(x, mr)). Let n be the maximal length of a big cylinder C containing both Ĉ and C′. Fix a cylinder C with this property. Then n < n′ and Ĉ and C′ are parts of big subcylinders of C of length n + 1 separated by a single gap D in C. From Lemma 4.1 we get |D| ≥ λ |C|. On the other hand, D ⊂ ω(I(x, mr)) implies c_{1}|D| ≤ 2mr, where c_{1} > 0 is one of the constants in (4.8). Hence

λ |C| ≤ 2mr/c_{1} . (5.2)

¿From the choice of C, it is clear that

ω(I(x, mr) ∩ Λ) ⊂ C. (5.3)

Finally, applying Corollary 4.3(a) to C′ and C, we get

|C′| ≤ A θ^{n′−n}|C|, (5.4) where θ = 1 − 2k_{0}λ′ ∈ (0, 1). This was derived from the assumption that ω(I(x, mr)) had a common point with another big cylinder of length n′. If this is not the case, then ω(I(x, mr) ∩ Λ) ⊂ C′, so taking C = C′ and therefore n = n′, we get (5.3) and (5.4) trivially satisfied. Moreover, it follows from C_{i_{0},j_{0}}[η_{1}, …, η_{n′}, η_{n′+1}] ⊂ ω(I(x, mr)), Lemma 4.1 and (4.8) that (5.2) also holds. Hence in the two possible cases there exists a big cylinder C of length n such that (5.2), (5.3) and (5.4) hold.

Next, we deal with ω(I(x, r)). Let C′′ = C_{i_{0},j_{0}}[ζ], ζ = (ζ_{1}, …, ζ_{n′′}), be a big cylinder of minimal length containing ω(x) and contained in ω(I(x, r)). Then |C′′′| ≥ r/c_{2}, where C′′′ = C_{i_{0},j_{0}}[ζ_{1}, …, ζ_{n′′−1}]. It follows from ω(x) ∈ C′′ ⊂ ω(I(x, r)) ⊂ ω(I(x, mr)) and ω(x) ∈ C′ that n′′ ≥ n′ and ζ_{j} = η_{j} for j = 1, …, n′. That is, C′′ ⊂ C′ ⊂ C. Now Corollary 4.3(a) implies |C′′| ≤ Aθ^{n′′−n}|C|. Combining the latter with |C′′| ≥ λ|C′′′| ≥ λr/c_{2} and (5.2), gives λr/c_{2} ≤ Aθ^{n′′−n} 2mr/(λc_{1}), which implies

n′′ − n ≤ \frac{log(2Amc_{2}/(λ^{2}c_{1}))}{log(1/θ)}. (5.5)

There exist a small cylinder C[ı] of minimal length with ω(x) ∈ ω(C[ı]) ⊂ C′′ and a small cylinder C[ȷ] of minimal length such that ω(C[ı]) ⊂ ω(C[ȷ]) ⊂ C. Denote by p′′ and p the lengths of ı and ȷ, respectively, and notice that p ≤ p′′ and p′′ − p ≤ (n′′ − n)/k. Notice that

g_{p′′}(x) = g(x) + g(σ(x)) + … + g(σ^{p′′−1}(x)) = g_{p}(x) + g_{p′′−p}(y),

where y = σ^{p}(x). Clearly, g_{p′′−p}(y) ≥ −(p′′ − p)g_{0}. Then (5.1) and (5.5) yield:

\frac{ν(C[ȷ])}{ν(C[ı])} ≤ \frac{βe^{pc}e^{g_{p}(x)}}{αe^{p′′c}e^{g_{p′′}(x)}} ≤ \frac{β}{α} · e^{−(p′′−p)c}e^{(p′′−p)g_{0}} ≤ \frac{β}{α} · e^{(p′′−p)|c−g_{0}|} ≤ \frac{β}{α} · e^{(n′′−n)|c−g_{0}|/k} ≤ D′_{m} ,

where D′_{m} = \frac{β}{α} · e^{\frac{|c−g_{0}| log(2Amc_{2}/(λ^{2}c_{1}))}{k log(1/θ)}} . Thus,

ν(C[ȷ]) ≤ D′_{m}ν(C[ı]) ≤ D′_{m}ν(I(x, r)). (5.6)

It remains to prove that ν(C) ≤ const ν(C[ȷ]). To do this, we first use Lemma 5.2 and represent C ∩ Λ as a finite disjoint union of images (under ω) of sets of the form C[ȷ′] ∩ Λ, where C[ȷ′] is a maximal small cylinder with C[ȷ′] ∩ Λ ≠ ∅ and C[ȷ′] ⊂ C. According to Lemma 5.3, the number of such cylinders is ≤ k^{2kℓ_{0}} and for each of them we have ν(C[ȷ′]) ≤ A_{2}ν(C[ȷ]). This, (5.3) and (5.6) imply

ν(I(x, mr)) ≤ ν(ω^{−1}(C) ∩ U) ≤ ∑_{ȷ′} ν(C[ȷ′]) ≤ A_{2}k^{2kℓ_{0}}ν(C[ȷ]) ≤ D′′_{m}ν(I(x, r)),

where D′′_{m} = D′_{m}A_{2}k^{2kℓ_{0}} > 0.

To deal with the case mr ≥ dist_{U′}(∂U′, U′ ∩ Λ), introduce the constant

r′ = min_{1≤i≤k} dist_{U_{i}}(∂U_{i}, U_{i} ∩ Λ) > 0.

We claim that there exists a constant D′′′_{m} > 0 such that for r′′ = r′/m we have

ν(I(x, r′′)) ≥ D′′′_{m} , x ∈ U ∩ Λ. (5.7)

To prove this we will essentially use some of the above arguments. Consider an arbitrary x ∈ U ∩ Λ, and let C′′ = C_{i_{0},j_{0}}[ζ] be a maximal big cylinder containing ω(x) and contained in ω(I(x, r′′)). If n′′ is the length of C′′, then as above for a big cylinder C′′′ of length n′′ − 1 with C′′ ⊂ C′′′ we get |C′′′| > r′′/c_{2}. This and Lemma 4.1 imply |C′′| ≥ λ|C′′′| ≥ λr′′/c_{2}. On the other hand, we have |C′′| < Eμ^{n′′} (cf. the proof of Lemma 5.3). Hence Eμ^{n′′} > λr′′/c_{2}

and therefore n′′ < \frac{log(Ec_{2}/(λr′′))}{log(1/μ)}. There exist a maximal small cylinder C[ı] with ω(x) ∈

ω(C[ı]) ⊂ C′′. If p′′ is its length, then we must have p′′ ≤ kn′′, so p′′ < \frac{k log \frac{Ec_{2}}{λr′′}}{log(1/μ)}. Now

C[ı] ⊂ I(x, r′′) and (5.1) yield

ν(I(x, r′′)) ≥ ν(C[ı]) ≥ αe^{p′′c}e^{g_{p′′}(x)} ≥ αe^{−p′′|c−g_{0}|} ≥ D′′′_{m} = αe^{−^{\frac{k|c−g_{0}| log \frac{Ec_{2}m}{λr′′}}{log(1/μ)}}} .

This proves (5.7).

Setting D_{m} = max{D′′_{m}, 1/D′′′_{m}}, it follows from the above that ν(I(x, mr)) ≤ D_{m}ν(I(x, r)) for any r > 0 and x ∈ U ∩ Λ.

6 The temporal distance function Δ

Recal the definition of the function Δ(x, y) from Sect. 2. This function has been naturally involved in various constructions in ergodic theory (cf. e.g. [Ra], [KB], [Ch1]). The following proposition is derived by adapting an argument of Katok and Burns [KB].

Proposition 6.1. For every σ ∈ Λ∩S_{ϵ}^{∗}(∂K) there exists δ > 0 such that for x ∈ W_{ϵ}^{u}(σ) and y ∈ W_{ϵ}^{s}(σ) ∩ Λ we have d_{x}Δ(x, y)_{| x=σ} ≠ 0 whenever 0 < dist(y, σ) < δ.

Proof. Consider the 1-form α = ∑_{j=1}^{2} ξ_{j}dx_{j} on T^{∗}(Ω), where x = (x_{1}, x_{2}) ∈ Ω, ξ = (ξ_{1}, ξ_{2}) ∈ IR^{2}. Notice that for σ = (x, ξ) ∈ T^{∗}(Ω), x being in the interior of Ω, and for any (y, η) ∈ T_{σ}(S^{∗}(Ω)) = T_{x}(Ω) × T_{ξ}(S^{1}) we have α_{σ}(y, η) = ⟨ξ, y⟩. Thus, α(X(σ)) = 1, where X(σ) = (ξ; 0) is the Hamilton vector field determined by the Hamilton function H(z, ζ) = \frac{1}{2}(∥ζ∥^{2} − 1) generating the billiard flow on S^{∗}(Ω). Also notice that for σ ∈ Λ we have

Ker(α_{σ}) = {(y, η) ∈ T_{σ}(S^{∗}(Ω)) : y ⊥ ξ} = E^{u}(σ) ⊕ E^{s}(σ) ,

where E^{u}(σ) and E^{s}(σ) are the tangent spaces at σ to W_{ϵ}^{u}(σ) and W_{ϵ}^{s}(σ), respectively. This follows from the geometric properties of the stable and unstable manifolds for dispersing billiards (cf. [Si2]). Since dα = ∑_{j=1}^{n} dξ_{j} ∧ dx_{j} is the restriction to S^{∗}(Ω) of the standard symplectic form on T^{∗}(Ω), it follows that for any v ∈ E^{u}(σ) \ {0} there exists w ∈ E^{s}(σ) with dα (v, w) ≠ 0.

Let σ ∈ Λ, v ∈ E^{u}(σ) and w ∈ E^{s}(σ) be such that ∥v∥ = ∥w∥ = 1. Let x(s), |s| ≤ δ, and y(t), |t| ≤ δ for some small δ > 0 be local C^{1} parametrizations of W_{ϵ}^{u}(σ) and W_{ϵ}^{s}(σ), respectively, with x(0) = y(0) = σ, ∥ẋ(s)∥ = ∥ẏ(t)∥ = 1 for all s and t and ẋ(0) = v, ẏ(0) = w. Using the argument from the proof of Lemma 3.3 in [KB], one gets

Δ(x(s), y(t)) = dα (sv, tw) + o(st)

as s → 0 and t → 0 with x(s), y(t) ∈ Λ. Since Δ(x(0), y(t)) = 0, this yields

\frac{d}{ds}Δ(x(s), y(t))_{|s=0} = tdα (v, w) + o(t).

Hence this derivative is non-zero whenever dα(v, w) ≠ 0 and t ≠ 0 is sufficiently small. It follows from dim E^{u}(σ) = dim E^{s}(σ) = 1 that v and w span these two subspaces. Clearly dα(v, w) ≠ 0; otherwise E^{u}(σ) ⊕ E^{s}(σ) ⊕ (IR · X(σ)) would be a 3-dimensional Lagrangean subspace of T_{σ}(Ω) for the symplectic form dα which is impossible. Finally, a simple geometric argument shows that 0 is a limit point of both {s ≠ 0 : x(s) ∈ Λ} and {t ≠ 0 : y(t) ∈ Λ}.

We are now going to use the Markov family {R_{i}}_{i=1}^{k} discussed in Sect. 2.

The following lemma is similar to Lemma 8 in [D]. The proof is also quite similar to that in [D]. We sketch it for completeness and because the objects constructed in its proof will be used later on.

Lemma 6.2. There exist constants δ_{0} > 0 and N_{0} ∈ IN, such that for any integer N ≥ N_{0} there exists an open neighbourhood U = U(N) of Λ ∩ U in U and C^{1} maps v_{1}, v_{2} : U ⟶ U such that σ^{N}(v_{i}(u)) = u (i = 1, 2) for u ∈ U, and

δ_{0} ≤ ∥d_{u} [τ_{N}(v_{1}(u)) − τ_{N}(v_{2}(u))]∥ ≤ Const , u ∈ U.

Proof. Set U^{(n)} = { P^{n}(x) : x ∈ U , P^{j}(x) exists for all j = 1, 2, …, n }. Notice that U ∩ Λ ⊂ U^{(n)} for all n > 1 (follows from the conjugacy between σ and the Bernoulli shift on Σ_{A}^{+}). For z, y ∈ R̃_{i} we will denote by d_{x}Δ(z, y) the differential of the function F^{u}(z) ∋ x ↦ Δ(x, y) ∈ IR at x = z. For x ∈ R_{1} denote F u ; R̃_{1}(x) = R̃_{1} ∩ pr_{D_{1}}(F^{u}(x)), where pr_{D_{1}} : ϕ_{[−ϵ,ϵ]}(D_{1}) ⟶ D_{1} is the natural projection along the orbits of ϕ_{t}. It follows from Proposition 6.1 that if z ∈ R_{1} and y ∈ W_{ϵ}^{s}(z) ∩ Λ \ {z} ⊂ R_{1} is sufficiently close to it, then d_{x}Δ(x, y)_{|x=z} ≠ 0. Since U^{(n)} fills in R_{1} densily as n → ∞, taking n_{0} large enough, we can find y_{0} ∈ R_{1} ∩ U^{(n_{0})} such that y_{0} ∉ U_{1} but y_{0} is close to U_{1}, so that the above remark involving Proposition 6.1 applies with y = y_{0} and z = u_{0} = p(y_{0}), where p : R̃_{1} ⟶ U_{1} is the projection along the leaves of F^{s}. That is, d_{u_{1}}Δ(u_{1}, y_{0})_{|u_{1}=u_{0}} ≠ 0, where u_{1} ∈ U_{1}. Let π : U_{1} ⟶ F u ; R̃_{1}(y_{0}) be the projection along the leaves of F^{s}. Clearly u_{0} ∈ Λ, so u_{0} ∈ R_{1}. Fix n_{0} > 1, y_{0} and u_{0} with this property.

In what folows u_{1} and u_{2} will be variables in U_{1}. Since

d_{u_{1}}Δ(u_{1}, π(u_{0}))_{u_{1}=u_{0}} = d_{u_{1}}Δ(u_{1}, y_{0})_{u_{1}=u_{0}} ≠ 0 ,

there exists an open neighbourhood U_{0} of u_{0} in U_{1} and δ′ ∈ (0, 1/3) such that

∥d_{u_{1}}Δ(u_{1}, π(u_{2}))∥ ≥ δ′ u_{1}, u_{2} ∈ U_{0} . (6.1)

Recall that y_{0} ∈ R_{1} ∩ U^{(n_{0})}, so F u ; R̃_{1}(y_{0}) ∩ U^{(n_{0})} ≠ ∅. Let O_{1} be a small open connected neighbourhood of y_{0} in F u ; R̃_{1}(y_{0}) ∩ U^{(n_{0})} and let f_{1} : O_{1} ⟶ U be an arbitrary C^{1} map such that P^{n_{0}}(f_{1}(x)) = x for all x ∈ O_{1}. Shrinking U_{0} if necessary, we may assume that π(U_{0}) ⊂ O_{1}. Setting w_{1}(z) = f_{1}(π(z)), z ∈ U_{0}, we get a C^{1} map w_{1} : U_{0} ⟶ U with P^{n_{0}}(w_{1}(z)) = π(z) and therefore σ^{n_{0}}(w_{1}(z)) = z for all z ∈ U_{0}. In the same way, using the fact that u_{0} ∈ U_{1} ∩ Λ ⊂ U^{(n_{0})}, one constructs a C^{1} map w_{2} : U_{0} ⟶ U (shrinking U_{0} again if necessary) with P^{n_{0}}(w_{2}(z)) = z for all z ∈ U_{0}. Then σ^{n_{0}}(w_{2}(z)) = z for all z ∈ U_{0}. Thus we got C^{1} maps

w_{i} : U_{0} ⟶ U , σ^{n_{0}}(w_{i}(z)) = z , z ∈ U_{0} (i = 1, 2). (6.2)

Set V_{i} = w_{i}(U_{0}), i = 1, 2. Notice that (6.2) and (2.1) imply c_{0}/γ_{1}^{n_{0}} ≤ ∥dw_{i}(z)∥ ≤ 1/(c_{0}γ^{n_{0}}) for all z ∈ U_{0} and i = 1, 2.

A property of Δ(x, y) which will be used next is the following ([D]): if x, y ∈ U ∩ Λ and the integer n > 0 are such that P^{n}(x) and P^{n}(y) are sufficiently close and therefore belong to the same rectangle R_{i}, then

τ_{n}(y) − τ_{n}(x) = Δ(P^{n}(x), P^{n}(y)) = Δ(p(P^{n}(x)), P^{n}(y)) ,

where as before p : R̃_{i} ⟶ U_{i} is the projection along the leaves of F^{s}. This follows from a simple geometric observation using the definition of Δ. Using this property and the choice of the maps w_{i}, for z ∈ U_{0} and z′ ∈ U_{0} close to z we get

[τ_{n_{0}}(w_{1}(z′)) − τ_{n_{0}}(w_{2}(z′))] − [τ_{n_{0}}(w_{1}(z)) − τ_{n_{0}}(w_{2}(z))]
= −Δ(P^{n_{0}}(w_{1}(z′)), P^{n_{0}}(w_{1}(z))) + Δ(P^{n_{0}}(w_{2}(z′)), P^{n_{0}}(w_{2}(z)))
= −Δ(π(z′), π(z)) + Δ(z′, z) = −Δ(z′, π(z)) + Δ(z, π(z)).

(Here we used the fact that Δ(z′, z) = Δ(z, π(z)) = 0.) Then (6.1) implies

∥d_{z}[τ_{n_{0}}(w_{1}(z)) − τ_{n_{0}}(w_{2}(z))]∥ ≥ δ′ (z ∈ U_{0}). (6.3)

Since U_{0} is open in U and U_{0} ∩ Λ ≠ ∅, it follows that if n_{1} > 0 is large enough, then U = σ^{n_{1}}(U_{0}) (cf. the Convention in Sect. 2). The existence of such n_{1} can be easily derived by the symbolic representation obtained for ϕ_{t} on Λ by means of the Markov family. Shrinking U_{0} if necessary (now the new U_{0} can no longer be connected; necessarily it will be the union of finitely many connected open sets the distance between any two of them being positive), we may assume that σ^{n_{1}}(x) is well-defined for all x ∈ U_{0} and

Λ ∩ U ⊂ σ^{n_{1}}(U_{0}). (6.4)

We will assume that n_{1} is so large that σ^{n_{1}} : U_{0} ⟶ U is expanding, i.e. γ_{1}^{n_{1}} > 1/c_{0} (cf. (2.1)). Fix U_{0} and n_{1} > 0 with the properties listed above. Set

U = σ^{n_{1}}(U_{0}).

Since σ^{n_{1}} : U_{0} ⟶ U is a C^{1} diffeomorphism, its inverse z(u) = (σ^{n_{1}})^{−1}(u) is a C^{1} diffeomorphism (6.5) z : U ⟶ U_{0} such that σ^{n_{1}}(z(u)) = u , u ∈ U .

This and (2.1) imply c_{0}/γ_{1}^{n_{1}} ≤ ∥dz(u)∥ ≤ 1/(c_{0}γ^{n_{1}}) for all u ∈ U.

Next, as in the proof of Lemma 9 in [D] one derives that there exists an integer n_{2} > 0 such that for n > n_{2} there exist C^{1} maps ṽ_{i} : V_{i} ⟶ U (i = 1, 2) with σ^{n}(ṽ_{i}(w)) = w for all w ∈ V_{i} and such that

∥d_{w}τ_{n}(ṽ_{i}(w))∥ ≤ \frac{c_{0}^{3}γ^{n_{0}+n_{1}} δ′}{4γ_{1}^{n_{1}}} (w ∈ V_{i}). (6.6)

Fix n_{2} > 0 and C^{1} maps ṽ_{i} : V_{i} ⟶ U with these properties. Set N_{0} = n_{0} + n_{1} + n_{2} and let N > N_{0}. Then n = N − n_{0} − n_{1} > n_{2}, so for each i = 1, 2 there exists a C^{1} diffeomorphism

ṽ_{i} : V_{i} ⟶ U with σ^{N−n_{0}−n_{1}}(ṽ_{i}(w)) = w , w ∈ V_{i} , (6.7)

such that (6.6) holds with n = N − n_{0} − n_{1} and δ′′ as above. Now define the C^{1} maps

v_{i} : U ⟶ U , v_{i}(u) = ṽ_{i}(w_{i}(z(u))). (6.8)

Then we have σ^{N−n_{1}}(v_{i}(u)) = σ^{n_{0}}(σ^{N−n_{0}−n_{1}}(v_{i}(u))) = σ^{n_{0}}(w_{i}(z(u))) = z(u) (which is the same for i = 1, 2) for all u ∈ U. Consequently, σ^{m}(v_{1}(u)) = σ^{m}(v_{2}(u)) for all m ≥ N − n_{1} and u ∈ U. Thus, τ_{N}(v_{1}(u)) − τ_{N}(v_{2}(u)) = τ_{N−n_{1}}(v_{1}(u)) − τ_{N−n_{1}}(v_{2}(u)). Using this, (6.6), (6.3), ∥d_{z}w_{i}(z)∥ ≤ 1/(c_{0}γ^{n_{0}}), and the Chain Rule, as in [D] one gets

∥d_{u} [τ_{N}(v_{1}(u)) − τ_{N}(v_{2}(u))] ∥ ≥ δ′ \frac{c_{0}}{γ_{1}^{n_{1}}} − 2δ′′ \frac{1}{c_{0}γ^{n_{0}}} \frac{1}{c_{0}γ^{n_{1}}} = δ′ \frac{c_{0}}{2γ_{1}^{n_{1}}}

for all u ∈ U. So, we can take δ_{0} = δ′ \frac{c_{0}}{2γ_{1}^{n_{1}}} . It is also clear that there exists a constant Const > 0 depending on n_{0} and n_{1} such that ∥d_{u} [τ_{N}(v_{1}(u)) − τ_{N}(v_{2}(u))] ∥ ≤ Const for all u ∈ U. This proves the assertion.

7 Proof of Lemma 3.4

¿From now until the end of the section we will use the open subset U_{0} of U, the integers n_{0}, n_{1}, n_{2} and N_{0} = n_{0} + n_{1} + n_{2}, the set U = σ^{n_{1}}(U_{0}), the number δ_{0} and the map z(u) with (6.5) constructed in the proof of Lemma 6.2. In this section we will first use the results from Sect. 4 to prove the existence of special partitions of U_{0} (and in fact any connected subcurve of U having common points with Λ).

Fix an arbitrary orientation on each connected component U_{i} of the U_{0}. A family {γ_{j}}_{j=1}^{q} of closed conected curves (intervals) in U_{0} will be called ordered if for every j = 1, …, q − 1 such that the curves γ_{j} and γ_{j+1} belong to the same connected component of U_{0} we have: γ_{j+1} is after γ_{j} (with respect to the positive direction along U_{0}) and γ_{j} ∩ γ_{j+1} consists of one point which belongs to ∂γ_{j} ∩ ∂γ_{j+1}. We will say that an ordered family {γ_{j}}_{j=1}^{q} of closed connected curves in U_{0} has the triple intersection property if for every j with γ_{j} ∩ Λ ≠ ∅ there exists j′ with |j − j′| ≤ 1 such that the curves γ_{j′}, γ_{j′+1}, γ_{j′+2} are contained in the same connected component of U and all of them have a common point with Λ.

Lemma 7.1. There exist constants 0 < c_{1} < C_{1} and μ_{0} ∈ (0, 1/2) such that for every real number b with |b| ≥ 1 and every sufficiently small ϵ_{1} > 0 there exist a finite ordered family Γ(b, ϵ_{1}) = {γ_{j}}_{j=1}^{q} (q ≥ 3k) of curves in U_{0} diffeomorphic to [0, 1] such that:

(a) c_{1} \frac{ϵ_{1}}{|b|} ≤ |γ_{j}| ≤ C_{1} \frac{ϵ_{1}}{|b|} for all j ;

(b) U_{0} = ∪_{j=1}^{q}γ_{j} and {γ_{j}}_{j=1}^{q} has the triple intersection property.

(c) dist_{γ_{j}}(∂γ_{j} , γ_{j} ∩ Λ) ≥ 2μ_{0}|γ_{j}| for all j with γ_{j} ∩ Λ ≠ ∅.

Proof. Let |b| ≥ 1 and let ϵ_{1} > 0 be such that

ϵ_{1} ≤ min \{ \frac{|U′|}{6} , dist_{U′}(∂U′, U′ ∩ Λ) \}

for any connected component U′ of U_{0}. It is enough to find a representation of each connected component of U_{0} as a finite union of subcurves with the properties described in Lemma 7.1.

Let U′ be an arbitrary connected component of U_{0}. Consider an arbitrary representation U′ = ∪_{i=1}^{p}I′_{i}, of U′ as a finite union of connected closed (i.e. containing their ends) curves

with disjoint interiors such that \frac{ϵ_{1}}{|b|} ≤ |I′_{i}| ≤ \frac{2ϵ_{1}}{|b|} for all 1 ≤ i ≤ p, and for each i < p,

I′_{i} and I′_{i+1} have a common end point. That is, each I′_{i} is a subcurve of U′ of the form I′_{i} = [x_{i−1}, x_{i}], where x_{0} and x_{p} are the initial and the terminal points of U′, respectively.

First, we will slightly change the points x_{i} (whenever necessary) to ensure that dist_{U′}(∂I′_{i}, I′_{i} ∩ Λ) is sufficiently large for each i. Fix i for a moment. If the interval I(x_{i}, ϵ_{1}/6|b|) contains no points of Λ, leave x_{i} unchanged. Assume that I(x_{i}, ϵ_{1}/6|b|)∩Λ ≠ ∅ and choose an arbitrary x′_{i} in this set. Applying Corollary 4.3(b) to I(x′_{i}, ϵ_{1}/6|b|), we get a sequence of subcurves Θ_{j} with the properties (i) and (ii). Take an arbitrary Θ_{j} which is a gap, i.e. its interior does not contain points of Λ, and replace x_{i} by the middle of Θ_{j}. In this way we get new intervals I′_{i} and I′_{i+1}. Repeating this procedure several times by induction starting from I′_{1}, we end up with a new family of intervals I′_{i} (we will use the same notation for them) such that

\frac{ϵ_{1}}{3|b|} ≤ |I′_{i}| ≤ \frac{5ϵ_{1}}{3|b|} , dist_{U_{0}}(∂I′_{i}, I′_{i} ∩ Λ) ≥ λ_{0} \frac{ϵ_{1}}{12|b|} (i = 1, …, p).

Fix for a moment i = 1, …, p with I′_{i} ∩ Λ ≠ ∅ and choose an arbitrary y_{i} ∈ I′_{i} ∩ Λ. Then for r = λ_{0} \frac{ϵ_{1}}{12|b|} we have I(y_{i}, r) ⊂ I′_{i}. Applying Corollary 4.3(b) to I(y_{i}, r), we get an ordered family {Θ_{j}}_{j=1}^{2k_{0}−1} of connected subcurves of I(y_{i}, r) having the properties listed in Corollary 4.3(b). First, consider the case k_{0} > 3. Denote by z_{j} the middle of the curve Θ_{2j+1} for 1 ≤ j ≤ k_{0} − 2, and let z_{0} = x_{i−1} and z_{k_{0}−1} = x_{i} (the initial and terminal points of I′_{i}). For j = 1, … k_{0} −1, let γ_{j}^{(i)} be the connected subcurve of I′_{i} with end points z_{j−1} and z_{j}. In this way we get a representation of I′_{i} as a union of k_{0} −1 closed connected subcurves γ_{j}^{(i)} with disjoint interiors and such that γ_{j}^{(i)} ∩ Λ ≠ ∅ and dist_{U_{0}}(∂γ_{j}^{(i)}, γ_{j}^{(i)} ∩ Λ) ≥ 2μ_{0} \frac{ϵ_{1}}{|b|} for all j = 1, …, k_{0} − 1, where μ_{0} = λ_{0}^{2}/48 is a positive global constant. Moreover, |γ_{j}^{(i)}| ≤ 2r = \frac{λ_{0}ϵ_{1}}{6|b|} and |γ_{j}^{(i)}| ≥ λ_{0}r = \frac{λ_{0}^{2}ϵ_{1}}{12|b|}. When k_{0} > 3 we have k_{0} − 1 ≥ 3, so the family

{γ_{j}^{(i)}}_{j=1}^{k_{0}−1} consists of at least three different curves. Thus, the family

{γ_{j}^{(i)} : j = 1, …, k_{0} − 1, i = 1, … p, I′_{i} ∩ Λ ≠ ∅} ∪ {I′_{i} : I′_{i} ∩ Λ = ∅}

has all the properties listed in Lemma 7.1 for the connected component U′ of U_{0}.

Next, consider the case k_{0} = 3. Then in the above procedure we only get two subcurves γ_{j}^{(i)} which is not enough. We need to apply Corollary 4.3(b) (or rather its proof) once again, say to each of the subcurves Θ_{2} and Θ_{4} (in fact it is enough to do this to one of them; the argument in the proof of Corollary 4.3(b) applies easily, since Θ_{2} and Θ_{4} are cylinders). In this way for i = 2, 4 we get a representation Θ_{i} = ∪_{r=1}^{5}Θ_{i,r} where Θ_{i,r} are closed connected subcurves of Θ_{i} with disjoint interiors such that |Θ_{i,r}| ≥ λ_{0}|Θ_{i}| ≥ λ_{0}^{3} \frac{ϵ_{1}}{12|b|} for all r = 1, …, 5 and Θ_{i,r} ∩ Λ ≠ ∅ iff r = 2 or r = 4. Now define the family of intervals {γ_{j}^{(i)}}_{j=1}^{4} in such a way that each γ_{j}^{(i)} contains one of the subcurves Θ_{i,r} with j, r ∈ {2, 4} and the end points of γ_{j}^{(i)} are formed by the end points of the interval I′_{i} and the middles of the gaps Θ_{2,3}, Θ_{3} and Θ_{4,3}. Then ∪_{i=1}^{p}{γ_{j}^{(i)}}_{j=1}^{4} is a family of subcurves of U′ having all the properties listed in Lemma 7.1, where this time μ_{0} = λ_{0}^{3}/48.

Let b ∈ IR be such that 1/|b|^{α} ≤ a′_{0}. Fix an arbitrary ϵ_{1} such that

(7.1) 0 < ϵ_{1} ≤ min \{ \frac{1}{32C_{0}C_{1}} , dist_{U′}(∂U′, U′ ∩ Λ) , \frac{|U′|}{6} \}

for any connected component U′ of U, and fix a family Γ(b, ϵ_{1}) = {γ_{j}}_{j=1}^{q} of curves in U_{0} having the properties listed in Lemma 7.1.

For each j = 1, …, q denote by γ̃_{j} the connected subcurve of γ_{j} consisting of all points z ∈ γ_{j} with dist_{γ_{j}}(∂γ_{j}, z) ≥ μ_{0}|γ_{j}|. Then condition (c) in Lemma 7.1 implies

γ_{j} ∩ Λ ⊂ γ̃_{j} , 1 ≤ j ≤ q .

Given j with γ_{j} ∩ Λ ≠ ∅ fix an arbitrary z_{j} ∈ γ_{j} ∩ Λ ⊂ γ̃_{j} ∩ Λ. It follows from (a) and (c) in Lemma 7.1 that I(z_{j}, μ_{0}|γ_{j}|) ⊂ γ̃_{j} and μ_{0}|γ_{j}| ≥ \frac{c_{1}μ_{0}ϵ_{1}}{|b|}. Thus,

I (z_{j}, \frac{c_{1}μ_{0}ϵ_{1}}{|b|} ) ⊂ γ̃_{j} ⊂ γ_{j} , j = 1, …, q. (7.2)

Consequently, there exists a C^{1} function η_{j} : γ_{j} ⟶ [0, 1] such that η_{j} = 0 on a neighbourhood of ∂γ_{j}, η_{j} = 1 on γ̃_{j}, and^{9}

∥d_{u}η_{j}(u)∥ ≤ \frac{4|b|}{c_{1}μ_{0}ϵ_{1}} , u ∈ γ_{j} . (7.3)

For each j with γ_{j} ∩ Λ ≠ ∅ fix a C^{1} function η_{j} with the above properties.

Given j = 1, …, q and i = 1, 2, denote

Y_{j}^{i} = w_{i}(γ_{j}) , Z_{j} = σ^{n_{1}}(γ_{j}) , X_{j}^{i} = {v_{i}(u) : u ∈ Z_{j}} .

It then follows that γ_{j} = z(Z_{j}), X_{j}^{i} = ṽ_{i}(Y_{j}^{i}), and U = ∪_{j=1}^{q}Z_{j}. The following diagram may be of some help in figuring out where these subsets belong.

9On the real line, given δ > 0, the function η : IR ⟶ [0, 1], defined by η(x) = 1 for x < 0, η(x) = (δ^{2} − x^{2})^{2}/δ^{4} for 0 ≤ x ≤ δ, η(x) = 0 for x > δ, has the desired properties.

U  ⟶^{z(u)} ; ⟵_{σ^{n_{1}}} U_{0} ⟶^{w_{i}} ; ⟵_{σ^{n_{0}}}  V_{i}  ⟶^{ṽ_{i}} ; ⟵_{σ^{N−n_{0}−n_{1}}}  U
⋃  ⋃  ⋃  ⋃
Z_{j} = σ^{n_{1}}(γ_{j}) ⟶^{z(u)} γ_{j} ⟶^{w_{i}} Y_{j}^{i} = w_{i}(γ_{j}) ⟶^{ṽ_{i}} X_{j}^{i} = ṽ_{i}(Y_{j}^{i}) = v_{i}(Z_{j})

Let J be a subset of the set

Ξ = { (i, j) : 1 ≤ i ≤ 2 , 1 ≤ j ≤ q , γ_{j} ∩ Λ ≠ ∅ }.

Given μ ∈ (0, 1), following [D], we define the function m = m_{μ,b,J} : U ⟶ [0, 1] by

m(v) = \{ \begin{matrix}1 − μη_{j}(z(σ^{N}(v)) , v ∈ X_{j}^{i} for some (i, j) ∈ J , \\ 1 otherwise . \end{matrix}

Notice that v ∈ X_{j}^{i} is equivalent to v = v_{i}(u) for some u ∈ Z_{j}; then σ^{N}(v) is well-defined and σ^{N}(v) = σ^{N}(v_{i}(u)) = u ∈ Z_{j}. Consequently, z(u) is well-defined and z(u) ∈ γ_{j}. Since the curves γ_{j} have no common interior points and η_{j} is zero near ∂γ_{j} = z(∂Z_{j}), the definition of m is correct and m is C^{1}. Notice that if z(σ^{N}(v)) ∈ γ̃_{j} for some (i, j) ∈ J, then η_{j}(z(σ^{N}(v))) = 1, so m(v) = 1 − μ. It is worth mentioning that for v = v_{i}(u) with u ∈ Z_{j} we have

σ^{N−n_{0}−n_{1}}(v) = σ^{N−n_{0}−n_{1}}(ṽ_{i}(w_{i}(z(u))) = w_{i}(z(u)),

so σ^{N−n_{1}}(v) = σ^{n_{0}}(w_{i}(z(u))) = z(u) = z(σ^{N}(v)). Thus, m(v) = 1 − μη_{j}(σ^{N−n_{1}}(v)) for v ∈ X_{j}^{i}. Finally, notice that the definition of m, (7.3) and (2.1) imply

∥d_{v}m(v)∥ ≤ μ\frac{4|b|γ_{1}^{N}}{c_{1}μ_{0}ϵ_{1}c_{0}} , v ∈ U . (7.4)

Next, define the operator N = N_{ab}^{(J,μ)} : C^{1}(U) ⟶ C^{1}(U) by

(N h) (u) = (M_{ab}^{N}(m · h)) (u) , u ∈ U .

In the rest of this section and the next one we will choose a number μ ∈ (0, 1), and a subset J of Λ such that the operator N = N_{ab}^{(J,μ)} has some special properties. Let us say in advance that the only condition for μ will be of the form μ ≤ const e^{−constN}, where const is a positive constant that does not depend on b and N.

The following lemma contains statements similar to Proposition 6 and Lemma 11 in [D]. The proof is almost the same (Lemma 2.2 is applied several times), so we omit it.

Lemma 7.2. If E > 0, 1/|b|^{α} ≤ a′_{0}, μ > 0 and N satisfy the inequalities

4A_{0} ≤ E , γ^{N} ≥ 4A_{0} , μ ≤ min \{ \frac{1}{2} , \frac{c_{0}c_{1}μ_{0}ϵ_{1}E}{8γ_{1}^{N}} \} , (7.5)

then the following hold:

(a) N h ∈ K_{E|b|} for any h ∈ K_{E|b|};

(b) For any h, H ∈ C^{1}(U) such that H ∈ K_{E|b|}, |h(u)| ≤ H(u) and ∥dh(u)∥ ≤ E|b|H(u) for any u ∈ U, we have ∥d_{u}(L_{ab}^{N} h)(u)∥ ≤ E|b|(N H)(u) for all u ∈ U.

In what follows for each integer m > 0, D_{m} will denote a fixed constant with the property described in Proposition 5.1.

Modifying a definition in [D], given r > 0 and m > 0, a subset W of U will be called (r, m)-dense in U if for every u ∈ Λ ∩ U there exists v ∈ Λ ∩ U such that I(v, r) ∩ U ⊂ W ∩ I(u, mr).

The following is a consequence of Proposition 5.1.

Proposition 7.3. Let E > 0, m > 0 and let ϵ = ϵ(m, E) = \frac{1}{5 e^{2mE}D_{m}}. Then for any

r > 0, any (r, m)-dense subset W of U and any H ∈ K_{E/r}, we have

∫_{W} H^{2} dν ≥ ϵ ∫_{U} H^{2} dν .

Proof. Since supp(ν) ⊂ Λ ∩ U = Λ ∩ U, in the above integrals it is enough to integrate over Λ ∩ W and Λ ∩ U, respectively.

Let r > 0, W be an (r, m)-dense subset of U and H ∈ K_{E/r}. Divide each connected component U_{i} of U into connected curves θ with disjoint interiors and such that mr/2 ≤ |θ| < mr. If mr ≤ |U_{i}| for some i, take θ = U_{i}. Let θ_{j} (j = 1, …, p) be the (finite) family of these curves that have non-empty intersection with Λ ∩ U. Thus, Λ ∩ U ⊂ ∪_{j=1}^{p}θ_{j}. Denote m_{j} = inf_{u∈θ_{j}∩U} H(u) and M_{j} = sup_{u∈θ_{j}∩U} H(u). Then (3.7) and H ∈ K_{E/r}(U) imply

H(u) ≥ m_{j} ≥ M_{j}e^{−Em} , u ∈ θ_{j} ∩ U. (7.6)

Next, for each j = 1, …, p choose an arbitrary point u_{j} ∈ θ_{j} ∩ Λ ∩ U. Then θ_{j} ⊂ I(u_{j}, mr). Different curves I(u_{i}, mr) and I(u_{j}, mr) may have common interior points, however every point of U belongs to the interior of at most 5 of these curves. Since u_{j} ∈ Λ ∩ U, it follows from the (r, m)-density of W that there exists v_{j} ∈ Λ ∩ U such that I(v_{j}, r) ∩ U ⊂ I(u_{j}, mr) ∩ W. Combining this with Proposition 5.1, gives

ν(I(u_{j}, mr) ∩ W) ≥ ν(I(v_{j}, r) ∩ U)) = ν(I(v_{j}, r)) ≥ \frac{1}{D_{m}} ν(I(u_{j}, mr)) ≥ \frac{1}{D_{m}} ν(θ_{j}).

This and (7.6) yield ∫_{W} H^{2}(u) dν(u) ≥ ϵ ∫_{U} H^{2}(u) dν(u), which proves the assertion.

Modifying a definition in [D], we will say that the subset J of Ξ is dense if for any j = 1, …, q with γ_{j} ∩ Λ ≠ ∅ there exists (i, j′) ∈ J such that |j − j′| ≤ 2. (Notice that (i, j′) ∈ J ⊂ Ξ implies γ_{j′} ∩ Λ ≠ ∅.)

Recall the positive constants a′_{0}, T and γ_{1} defined in Sect. 2. Taking a smaller a′_{0} > 0 if necessary, we may assume that there exists A_{2} > 0 such that ∥f^{(ab)} − f∥ ≤ a′_{0}A_{2} and ∥f^{(ab)} − (f − P τ)∥ ≤ A_{2} for |a| ≤ a′_{0} and 1/|b|^{α} ≤ a′_{0}. Fix a constant A_{2} > 0 with this property.

The following lemma is the analogue of Lemma 12 in [D].

Lemma 7.4. Given the numbers E and μ, there exist a_{0} = a_{0}(E, μ, N) > 0 and ϵ_{2} = ϵ_{2}(E, μ, N) such that

∫_{U}(N H)^{2}dν ≤ (1 − ϵ_{2}) ∫_{U} H^{2}dν

whenever J is dense, H ∈ K_{E|b|}, |a| ≤ a_{0}, and 1/|b|^{α} ≤ a_{0}. More precisely, we can take

a_{0} = min \{ (a′_{0})^{1/α} , −\frac{ln(1 − ϵ′μe^{−NT})}{2A_{2}N} \} , ϵ_{2} = \frac{ϵ′μe^{−NT}}{2} ,

where ϵ′ = \frac{1}{5D_{m′}e^{2m′Ec_{0}c_{1}ϵ_{1}μ_{0}γ^{n_{1}}}} and m′ = \frac{3C_{1}γ_{1}^{n_{1}}}{c_{0}^{2}c_{1}μ_{0}γ^{n_{1}}} .

Sketch of Proof. The definition of N and the Cauchy-Schwartz inequality imply

(N H)^{2}(u) = (M_{ab}^{N}(mH))^{2}(u) ≤ (M_{ab}^{N}m^{2})(u) · (M_{ab}^{N}H^{2})(u) (7.7)

for all u ∈ U, where m = m_{μ,b,J}.

Denote by W the set of those u ∈ U such that there exists (i, j) ∈ J with z(u) ∈ γ̃_{j}. According to the definition of m and the remark just after it, for such u we have m(v_{i}(u)) = 1 − μ. Using the density of J and property (a) in Lemma 7.1, one derives that

W is (r, m′)-dense in U, where r = \frac{μ_{0}c_{0}c_{1}ϵ_{1}γ^{n_{1}}}{|b|}.

Setting E′ = Eμ_{0}c_{0}c_{1}ϵ_{1}γ^{n_{1}}, we have H ∈ K_{E|b|} = K_{\frac{E′}{r}}. Notice that ϵ′ = \frac{1}{5e^{2m′E′}D_{m′}}, so it coincides with ϵ(m, E) defined in Proposition 7.3 with m and E replaced by m′ and E′. It then follows from Proposition 7.3 that ∫_{W} H^{2} dν ≥ ϵ′ ∫_{U} H^{2} dν. Since M_{ab}^{N}(H^{2}) = h^{2} for

some C^{1} positive function h on U, it follows that

∫_{W} M_{ab}^{N}(H^{2}) dν ≥ ϵ′ ∫_{U} M_{ab}^{N}(H^{2}) dν. (7.8)

Next, m^{2}(v) ≤ 1 for all v ∈ U implies M_{ab}^{N}(m^{2}) ≤ M_{ab}^{N}1 = 1. Moreover,

(M_{ab}^{N}(m^{2}))(u) ≤ 1 − μe^{−NT} , u ∈ W . (7.9)

To prove this, consider an arbitrary u ∈ W. It follows from the definition of W that there exists i = 1, 2 such that m(v_{i}(u)) = 1 − μ. Hence

(M_{ab}^{N}(m^{2}))(u) = ∑_{σ^{N} v=u} e^{f_{N}^{(ab)}(v)}m^{2}(v) ≤ ∑_{σ^{N} v=u,v≠v_{i}(u)} e^{f_{N}^{(ab)}(v)} + (1 − μ)e^{f_{N}^{(ab)}(v_{i}(u))}
≤ ∑_{σ^{N} v=u} e^{f_{N}^{(ab)}(v)} − μe^{f_{N}^{(ab)}(v_{i}(u))} ≤ (M_{ab}^{N} 1)(u) − μe^{−NT} = 1 − μe^{−NT} .

This proves (7.9). Using (7.7), (7.8) and (7.9), as in [D], we get the estimate

∫_{U}(N H)^{2} dν ≤ (1 − ϵ′μe^{−NT}) ∫_{U} M_{ab}^{N}(H^{2}) dν . (7.10)

As in the proof of Proposition 3.3, for |a| ≤ a_{0} we have

M_{ab}^{N}(H^{2})(u) = L_{f−P τ}^{N} (e^{(f^{(ab)}−f+P τ)_{N}} H^{2}) (u) ,

and e^{(f^{(ab)}−f^{(0)})_{N}} ≤ e^{a_{0}NT}. Using this and the choice of the constant A_{2} (cf. the text before the statement of Lemma 7.4), we get

∫_{U} M_{ab}^{N}(H^{2}) dν = ∫_{U} L_{f−P τ}^{N} (e^{(f^{(ab)}−f+P τ)_{N}} H^{2}) dν ≤ e^{a_{0}A_{2}N} ∫_{U} L_{f−P τ}^{N}(H^{2}) dν = e^{a_{0}A_{2}N} ∫_{U} H^{2} dν .

This and (7.10) yield ∫_{U}(N H)^{2} dν ≤ (1−ϵ_{2}) ∫_{U} H^{2} dν, since (1−ϵ′μe^{−NT})e^{a_{0}A_{2}N} ≤ 1−ϵ_{2}.

In what follows we assume that h, H ∈ C^{1}(U) are such that

H ∈ K_{E|b|} , |h(u)| ≤ H(u) , ∥dh(u)∥ ≤ E|b|H(u) for all u ∈ U . (7.11)

Given μ ∈ (0, 1/2), following [D], define the functions χ_{μ}^{(i)} : U ⟶ C (i = 1, 2) by

χ_{μ}^{(1)}(u) = \frac{|e^{(f_{N}^{(ab)}+ibτ_{N})(v_{1}(u))}h(v_{1}(u)) + e^{(f_{N}^{(ab)}+ibτ_{N})(v_{2}(u))}h(v_{2}(u))|}{(1 − 2μ)e^{f_{N}^{(ab)}(v_{1}(u))}H(v_{1}(u)) + e^{f_{N}^{(ab)}(v_{2}(u))}H(v_{2}(u))} ,

χ_{μ}^{(2)}(u) = \frac{|e^{(f_{N}^{(ab)}+ibτ_{N})(v_{1}(u))}h(v_{1}(u)) + e^{(f_{N}^{(ab)}+ibτ_{N})(v_{2}(u))}h(v_{2}(u))|}{e^{f_{N}^{(ab)}(v_{1}(u))}H(v_{1}(u)) + (1 − 2μ)e^{f_{N}^{(ab)}(v_{2}(u))}H(v_{2}(u))} .

The following lemma is analogous to Lemma 13 in [D].

Lemma 7.5. If ϵ_{1} > 0 and μ > 0 are sufficiently small, then for any j = 1, …, q with γ_{j} ∩ Λ ≠ ∅ there exist j′ = 1, …, q with |j′ − j| ≤ 2 and γ_{j′} ∩ Λ ≠ ∅ and i = 1, 2 such that

χ_{μ}^{(i)}(u) ≤ 1 , u ∈ Z_{j′} . (7.12)

We will prove this lemma first and then show how to derive Lemma 3.4 from it. The following lemma coincides with Lemma 14 in [D] and we omit its proof.

Lemma 7.6. If h and H satisfy (7.11) and

0 < ϵ_{1} ≤ \frac{1}{8EC_{1}} , (7.13)

then for all j = 1, …, q and i = 1, 2 we have:

(a) \frac{1}{2} ≤ \frac{H(v_{i}(u′))}{H(v_{i}(u′′))} ≤ 2 for all u′, u′′ ∈ Z_{j};

(b) Either for all u ∈ Z_{j} we have |h(v_{i}(u))| ≤ \frac{3}{4}H(v_{i}(u)), or |h(v_{i}(u))| ≥ \frac{1}{4}H(v_{i}(u)) for all u ∈ Z_{j}.

As in [D] (cf. Lemma 15 there), using Lemma 6.2, one derives the following

Lemma 7.7. Let ψ(u) = b[τ_{N}(v_{1}(u)) − τ_{N}(v_{2}(u))] for u ∈ U, and let j, j′ ∈ {1, 2, …, q} be such that |j − j′| = 2 and γ_{j} and γ_{j′} belong to the same connected component of U_{0}. Then c_{2}ϵ_{1} ≤ |ψ(u) − ψ(u′)| for u ∈ Z_{j} and u ∈ Z_{j′}, where c_{2} = δ_{0}c_{0}c_{1}γ^{n_{1}}.

Proof of Lemma 7.5. Assume that (7.13) holds and γ_{j} ∩ Λ ≠ ∅. It follows from the triple intersection property of the family {γ_{i}} that three consecutive intervals γ_{i} including γ_{j} belong to the same connected component of U_{0} and have common points with Λ. Suppose that γ_{j}, γ_{j+1}, γ_{j+2} is such a triple; the other cases are similar. Set

Z̃_{j} = Z_{j} ∪ Z_{j+1} ∪ Z_{j+2} .

If there exist j′ ∈ {j, j + 1, j + 2} and i = 1, 2 such that the first alternative in Lemma 7.6 (b) holds for Z_{j′} and i, then χ_{μ}^{(i)}(u) ≤ 1 for any u ∈ Z_{j′} follows immediately.

Assume that for every j′ ∈ {j, j + 1, j + 2} and every i = 1, 2 the second alternative in Lemma 7.6 (b) holds for Z_{j′} and i. As in [D], consider the function

Ψ(u) = Arg (e^{ibτ_{N} (v_{1}(u))}h(v_{1}(u))) − Arg (e^{ibτ_{N} (v_{2}(u))}h(v_{2}(u))) .

Since |h(v_{i}(u))| ≥ \frac{1}{4}H(v_{i}(u)) > 0, we can choose real continuous functions θ_{i}(u), u ∈ Z̃_{j}, i = 1, 2, such that h(v_{i}(u)) = e^{iθ_{i}(u)}|h(v_{i}(u))| for all u ∈ Z̃_{j}. Then ln h(v_{i}(u)) is well-defined with Im(ln(h(v_{i}(u))) = θ_{i}(u). Using the second alternative in Lemma 7.6(b), Lemma 2.1, σ^{N}(v_{i}(u)) = u and (7.11), we get

∥d_{u} ln h(v_{i}(u))∥ ≤ \frac{∥d_{v}h(v)∥}{|h(v_{i}(u))|} ∥dv_{i}(u))∥ ≤ \frac{4E|b|}{c_{0}γ^{N}}

for u ∈ Z̃_{j}. In particular, for u, u′ ∈ Z̃_{j} we have

| ln h(v_{i}(u)) − ln h(v_{i}(u′))| ≤ \frac{4E|b|}{c_{0}γ^{N}}dist_{U}(u, u′) ≤ \frac{4E|b|}{c_{0}γ^{N}} · \frac{3C_{1}γ_{1}^{n_{1}}ϵ_{1}}{c_{0}|b|} = \frac{C_{2}Eϵ_{1}}{γ^{N}} ,

where C_{2} = \frac{12C_{1}γ_{1}^{n_{1}}}{c_{0}^{2}} > 0 is a constant. This yields

|θ_{i}(u) − θ_{i}(u′)| ≤ | ln h(v_{i}(u)) − ln h(v_{i}(u′))| ≤ \frac{C_{2}Eϵ_{1}}{γ^{N}} , u, u′ ∈ Z̃_{j} . (7.14)

Let u ∈ Z_{j} and u′ ∈ Z_{j+2}. It follows from Lemma 7.7 and (7.14) that

|Ψ(u) − Ψ(u′)| ≥ |ψ(u) − ψ(u′)| − |θ_{1}(u) − θ_{1}(u′)| − |θ_{2}(u) − θ_{2}(u′)|
≥ c_{2}ϵ_{1} − 2C_{2} \frac{Eϵ_{1}}{γ^{N}} ≥ \frac{c_{2}}{2} ϵ_{1},

provided

4\frac{C_{2}}{c_{2}} E ≤ γ^{N} . (7.15)

Thus, assuming (7.15), we have |Ψ(u) − Ψ(u′)| ≥ \frac{c_{2}}{2} ϵ_{1} for all u ∈ Z_{j} and u′ ∈ Z_{j+2}. Hence either |Ψ(u)| ≥ \frac{c_{2}}{4} ϵ_{1} for all u ∈ Z_{j} or |Ψ(u′)| ≥ \frac{c_{2}}{4} ϵ_{1} for all u′ ∈ Z_{j+2}.

Assume for example that |Ψ(u)| ≥ \frac{c_{2}}{4} ϵ_{1} for all u ∈ Z_{j}. We will show that χ_{μ}^{(i)}(u) ≤ 1 for all u ∈ Z_{j}, i = 1, 2. Given u ∈ Z_{j}, consider the points

ρ_{1} = e^{(f_{N}^{(ab)}+ibτ_{N})(v_{1}(u))}h(v_{1}(u)) , ρ_{2} = e^{(f_{N}^{(ab)}+ibτ_{N})(v_{2}(u))}h(v_{2}(u))

in the complex plane C. Then |Arg(ρ_{1}) − Arg(ρ_{2})| = |Ψ(u)| ≥ ϵ = \frac{c_{2}}{4} ϵ_{1}, while the second alternative in Lemma 7.6 (b) (which holds by assumption) and Lemma 7.6(a) imply

\frac{|ρ_{1}|}{|ρ_{2}|} = e^{f_{N}^{(ab)}(v_{1}(u))−f_{N}^{(ab)}(v_{2}(u))} \frac{|h(v_{1}(u))|}{|h(v_{2}(u))|} ≤ ω = 8e^{a_{0}A_{3}N} ≤ 8e^{A_{3}N} ,

for some constant A_{3} > 0 (as in the proof of Lemma 7.4). By symmetry \frac{|ρ_{2}|}{|ρ_{1}|} ≤ ω. This yields |ρ_{1} + ρ_{2}| ≤ min( |ρ_{1}| + (1 − δ)|ρ_{2}| , (1 − δ)|ρ_{1}| + |ρ_{2}| ) , (7.16)

where

δ = \frac{2}{ω} sin^{2} \frac{ϵ}{4} = \frac{1}{4e^{A_{3}N}} sin^{2} (\frac{c_{2}ϵ_{1}}{16} ) .

Indeed, assume that |ρ_{1}| ≤ |ρ_{2}|; the other case is similar. Then \frac{|ρ_{2}|}{ω} ≤ |ρ_{1}| ≤ |ρ_{2}|, and we have to show that |ρ_{1} + ρ_{2}| ≤ |ρ_{1}| + (1 − δ)|ρ_{2}|. Clearly the angle α between the vectors ρ_{1} and ρ_{2} is α = |Arg(ρ_{1}) − Arg(ρ_{2})| ≥ ϵ. Consider the triangle with vertices A = 0, B = ρ_{1}, C = ρ_{1} + ρ_{2}. Then ^{}A̸BC = π − α ≤ π − ϵ. Moreover, \frac{|BC|}{ω} ≤ |AB| ≤ |BC|, ω ≥ 1. Then it follows from elementary geometry (apply the sine theorem twice) that

|AC| ≤ |AB| + (1 − \frac{2}{ω} sin^{2} \frac{ϵ}{4}) |BC|. That is,

|ρ_{1} + ρ_{2}| = |AC| ≤ |AB| + (1 − \frac{2}{ω} sin^{2} \frac{ϵ}{4}) |BC| = |ρ_{1}| + (1 − δ)|ρ_{2}|.

This proves (7.16). Choosing μ ≤ \frac{δ}{2} = \frac{1}{ω} sin^{2} \frac{ϵ}{4}, i.e.

μ ≤ \frac{1}{8e^{A_{3}N}} sin^{2} (\frac{c_{0}ϵ_{1}}{16} ) , (7.17)

we get χ_{μ}^{(2)}(u) ≤ 1 for all u ∈ Z_{j}. In the same way, using |ρ_{1} + ρ_{2}| ≤ (1 − δ)|ρ_{1}| + |ρ_{2}|, one gets χ_{μ}^{(1)}(u) ≤ 1 for all u ∈ Z_{j}. This proves the lemma.

Proof of Lemma 3.4. Take E = 4A_{0}, and fix a positive integer N ≥ n_{0} + n_{1} + n_{2} such that

γ^{N} ≥ max \{ 4A_{0} , 48C_{2}A_{0} , 16\frac{C_{2}A_{0}}{c_{2}} \} . (7.18)

Take ϵ_{1} > 0 with (7.1) and (7.13). Then fix an arbitrary μ = μ(N) with

0 < μ ≤ min \{ \frac{1}{2} , \frac{c_{0}c_{1}μ_{0}ϵ_{1}A_{0}}{2γ_{1}^{N}} , \frac{1}{8e^{A_{3}N}} sin^{2} (\frac{c_{0}ϵ_{1}}{16} ) \} .

Notice that with this choice of E, N, ϵ_{1} and μ, the inequalities (7.5), (7.13), (7.15) and (7.17) are satisfied. Finally, define a_{0} = a_{0}(μ, N) > 0 and ϵ_{2} = ϵ_{2}(μ, N) > 0 as in Lemma 7.4. Then choose U_{0}, z(u), v_{i}(u), etc. as in the proof of Lemma 6.2 and a family {γ_{j}} of subcurves of U_{0} as in the beginning of Sect. 7.

Next, denote by J the set of those (i, j) ∈ Ξ such that χ_{μ}^{(i)}(u) ≤ 1 for all u ∈ Z_{j}. It follows from Lemma 7.5 (cf. also its proof) that J is dense. Consider the operator N = N_{ab}^{J,μ} : C^{1}(U) ⟶ C^{1}(U), where |a| ≤ a_{0}, and 1/|b| ≤ a_{0}. Then Lemma 7.2 (a) implies property (a) in Lemma 3.4, while Lemma 7.4 gives property (b) in Lemma 3.4 with ρ = 1 − ϵ_{2} = ρ(μ, N).

To check (c) in Lemma 3.4, assume that h and H satisfy (7.11). Then Lemma 7.2 (b) implies ∥d_{u}(L_{ab}^{N}h)(u)∥ ≤ E|b|(N H)(u) for all u ∈ U. So, it remains to show that

|(L_{ab}^{N}h)(u)| ≤ (N H)(u) , u ∈ U. (7.19)

Let u ∈ U and let m = m_{μ,b,J}. If u does not belong to the interior Int(Z_{j}) of Z_{j} for any (i, j) ∈ J, then m(v) = 1 whenever σ^{N}v = u (since v ∈ X_{j}^{i} implies u = σ^{N}v ∈ Z_{j}). Hence

|(L_{ab}^{N}h)(u)| = | ∑_{σ^{N} v=u} e^{(f_{N}^{(ab)}+ibτ_{N})(v)}h(v)| ≤ (M_{ab}^{N}(mH))(u) = (N H)(u).

Assume that u ∈ Int(Z_{j}) for some (i, j) ∈ J; then u ∈ U. Without loss of generality we may assume that (1, j) ∈ J (it may happen that both (1, j) and (2, j) belong to J). Then χ_{μ}^{(1)}(u) ≤ 1. Notice that if σ^{N}v = u for some v and m(v) < 1, then we must have v = v_{i′}(u′) ∈ Int(X_{j′}^{i′}) for some (i′, j′) ∈ J. Then u = σ^{N}v = σ^{N}(v_{i′}(u′)) = u′ shows that j′ = j (since Int(Z_{j}) ∩ Int(Z_{j′}) = ∅ for j ≠ j′). Thus, σ^{N}v = u and m(v) < 1 imply v = v_{1}(u) or v = v_{2}(u).

Using χ_{μ}^{(1)}(u) ≤ 1, we get

|(L_{ab}^{N}h)(u)| ≤ |  ∑_{σ^{N} v=u, v≠v_{1}(u),v_{2}(u)} e^{(f_{N}^{(ab)}+ibτ_{N})(v)}h(v)|
+ |e^{(f_{N}^{(ab)}+ibτ_{N})(v_{1}(u))}h(v_{1}(u)) + e^{(f_{N}^{(ab)}+ibτ_{N})(v_{2}(u))}h(v_{2}(u))|
≤ ∑_{σ^{N} v=u, v≠v_{1}(u),v_{2}(u)} e^{f_{N}^{(ab)}(v)}|h(v)|

+ [(1 − 2μ)e^{f_{N}^{(ab)}(v_{1}(u))}H(v_{1}(u)) + e^{f_{N}^{(ab)}(v_{2}(u))}H(v_{2}(u))] . (7.20)

First, consider the case when (1, j) ∈ J but (2, j) ∉ J. Then amongst the v’s with σ^{N}v = u there is only one for which m(v) < 1; this is v = v_{1}(u) for which we have m(v_{1}(u)) ≥ 1 − μ > 1 − 2μ. Thus, (7.20) implies

|(L_{ab}^{N}h)(u)| ≤ ∑_{σ^{N} v=u, v≠v_{1}(u),v_{2}(u)} e^{f_{N}^{(ab)}(v)}m(v)H(v)|

+ [e^{f_{N}^{(ab)}(v_{1}(u))}m(v_{1}(u))H(v_{1}(u)) + e^{f_{N}^{(ab)}(v_{2}(u))}m(v_{2}(u))H(v_{2}(u))] = (N H)(u).

So, (7.19) holds in this case.

Next, assume that both (1, j) and (2, j) belong to J. Then both inequalities χ_{μ}^{(1)}(u) ≤ 1 and χ_{μ}^{(2)}(u) ≤ 1 hold. So, apart from (7.20), we also have

|(L_{ab}^{N}h)(u)| ≤ ∑_{σ^{N} v=u, v≠v_{1}(u),v_{2}(u)} e^{f_{N}^{(ab)}(v)}|h(v)|+[e^{f_{N}^{(ab)}(v_{1}(u))}H(v_{1}(u)) + (1 − 2μ)e^{f_{N}^{(ab)}(v_{2}(u))}H(v_{2}(u))] .

Taking the average of this and (7.20), one gets |(L_{ab}^{N}h)(u)| ≤ (N H)(u). Thus, (7.19) holds in all possible cases. This completes the proof of Lemma 3.4.

8 Exponential decay of correlations

In this section we very briefly describe the procedure from [D] that uses Corollary 3.3 and leads to the proof of Theorem 1.2. It works in exactly the same way, since it is practically carried out on the symbol space Σ_{A}. Therefore we omit the details.

Let F ∈ F_{α}(Λ) and let f ∈ F_{α}(U ∩ Λ) be defined by f(x) = ∫_{0}^{τ(x)} F (ϕ_{s}(x)) ds (cf. Sect. 2). Given continuous functions A and B on Λ, let

ρ_{A,B}(t) = ∫_{Λ} A(x)B(ϕ_{t}(x)) dν_{F} (x)

be the corresponding correlation function, where as in Theorem 1.2, ν_{F} is the Gibbs measure on Λ determined by F. One also needs the Laplace transform ρ̂_{A,B}(z) = ∫_{0}^{∞} ρ_{A,B}(t)e^{−zt} dt of ρ_{A,B}. Here z = a + ib is a complex variable. As before, we will denote by ν the Gibbs measure on U (with support in Λ ∩ U) determined by f − P τ.

The first step comprises Propositions 3 and 4 in [D] and is based on previous arguments of Pollicott [Po] and Ruelle [R1].

Step 1. There exist constants ϵ_{3} > 0, θ ∈ (0, 1), M > 0 and C_{3} > 0 such that for every z = a + ib with |a| ≤ ϵ_{3} and 1/|b| ≤ ϵ_{3} and every integer n ≥ 0 there exist linear operators Q_{n}(z), R_{n}(z) : F_{α}(Λ) ⟶ F_{α}(Λ ∩ U) such that:

(a) ∥Q_{n}(z)A∥_{0} ≤ C_{3}θ^{n}|b| ∥A∥_{α} and ∥R_{n}(z)A∥_{0} ≤ C_{3}θ^{n}|b| ∥A∥_{α} for all A ∈ F_{α}(Λ);

(b) ∥Q_{n}(z)A∥_{α} ≤ C_{3}M^{n}|b|^{2} ∥A∥_{α} and ∥R_{n}(z)A∥_{α} ≤ C_{3}M^{n}|b|^{2} ∥A∥_{α} for all A ∈ F_{α}(Λ);

(c) The function

\frac{1}{∥A∥_{0}∥B∥_{0}} [ρ̂_{A,B}(z) − ∑_{i,j=0}^{∞} ∫_{Λ∩U} (L_{f−(P +z)τ}^{i+j}(1 − L_{f−(P +z)τ})^{−1}[Q_{i}(z)A]) [R_{j}(z)B] dν]

is uniformly bounded.

The next step is given by Corollary 3 (b) in [D] (cf. also Lemma 18 there).

Step 2. For every β ∈ (0, α) there exists a constant C_{4} > 0 such that for z = a+ib with |a| ≤ ϵ_{3} and 1/|b| ≤ ϵ_{3} we have ∥(1−L_{f−(P +z)τ})^{−1}h∥_{0} ≤ C_{4} ln |b| ∥h∥_{β} for all h ∈ F_{β}(Λ∩U).

Then Corollary 4 in [D] gives

Step 3. For all β ∈ (0, α) and A, B ∈ F_{β}(Λ) there exists ϵ_{3} > 0 such that ρ̂_{A,B}(z) is analytic in the region |a| ≤ ϵ_{3}, 1/|b| ≤ ϵ_{3}. Moreover, |ρ̂_{A,B}(z)| ≤ C_{5}|b|^{2} ln |b| ∥A∥_{β} ∥B∥_{β} for some constant C_{5} > 0.

As in [D], this yields the exponential decay of the function ρ_{A,B}(t)−(∫_{Λ} A dν_{F})(∫_{Λ} B dν_{F}).

Appendix

Sketch of Proof of Lemma 4.2. We will use the following simple fact (cf. e.g. Lemma 3.7 in [I1]).

Sublemma. Let W′ and W′′ be the boundaries of two disjoint compact strictly convex domains in IR^{2}. Suppose W′ and W′′ are C^{4}-smooth and denote by ν′ and ν′′, respectively, their outer unit normal fields. Given x ∈ W′, let x(t) (t ≥ 0) be the shift of x along the billiard trajectory in the exterior of W′′ issued from (x, ν′(x)). Suppose x, y ∈ W′ are such that the trajectories {x(t) : t ≥ 0} and {y(t) : t ≥ 0} hit transversally W′′ at x(τ_{1}) and y(τ_{2}), respectively. Denote by k the curvature of W′ at x. Assuming that the angle φ(x) between the reflected trajectory {x(t) : t ≥ τ_{1}} and the unit normal ν′′(x(τ_{1})) is bounded from above by a constant φ_{0} ∈ (0, π/2), we have

(1 + τ_{1}k)∥x − y∥ − c∥x − y∥^{2} ≤ ∥x(τ_{i}) − y(τ_{i})∥ ≤ (1 + τ_{1}k)∥x − y∥ + C∥x − y∥^{2}

for i = 1, 2, where c > 0 and C > 0 are constants depending only on the minimum and maximum of the curvatures of W′ and W′′ and on the constant φ_{0}. Moreover, choosing c and C appropriately, for i = 1, 2 we have c∥x(τ_{i}) − y(τ_{i})∥ ≤ ∥x(τ_{1}) − y(τ_{2})∥ ≤ C∥x(τ_{i}) − y(τ_{i})∥ .

We now proceed with the proof of Lemma 4.2. For s ∈ [0, 2] and t ≥ 0, denote by u_{t}(s) the shift of (q(s), ν(q(s))) along the forward billiard trajectory γ(s) after time t and by v_{t}(s) ∈ S^{1} the direction of γ(s) at u_{t}(s). Let t_{0}(s) = 0 and let t_{1}(s) < … < t_{m}(s) be the times of the consecutive reflections of the trajectory γ(s) at ∂K. Then t_{j}(s) = d_{0}(s) + d_{1}(s) + … + d_{j−1}(s) and q_{j}(s) = u_{t_{j}}(s) for j = 1, …, n. Set t_{n+1}(s) = t_{n}(s) + d_{n}(s). For t ≠ t_{j}(s) (j = 1, 2 …, n), locally near u_{t}(s), X_{t} = {u_{t}(s) : s ∈ [0, 2]} is a smooth convex curve with ”outward” normal field {v_{t}(s) : s ∈ [0, 2]}. Denote by k_{t}(s) the curvature of X_{t} at u_{t}(s). It follows from old results in the theory of billiards (cf. e.g. [Si2]; see also the Appendix of [St1]) that for t = t_{j}(s) for some j we have k_{j}(s) = lim_{t↓t_{j}(s)} k_{t}(s), where

k_{j}(s) is defined by (4.4). Moreover, for t_{j}(s) < t < t_{j+1}(s), the curvature of X_{t} at u_{t}(s) is

k_{t}(s) = \frac{k_{j}(s)}{1 + (t − t_{j}(s))k_{j}(s)}.

Fix for a moment an arbitrary s_{0} ∈ (0, 1) ∪ (1, 2) and an arbitrary j = 1, …, n. Given s close to s_{0}, assume t_{j}(s_{0}) ≥ t_{j}(s) and t_{j+1}(s_{0}) ≤ t_{j+1}(s); the other cases are similar. Applying the Sublemma to x = u_{t_{j}(s_{0})}(s_{0}) = q_{j}(s_{0}) and y = u_{t_{j}(s_{0})}(s) with W′ = X_{t_{j}(s_{0})} and W′′ = ∂K near u_{t_{j+1}(s_{0})}(s_{0}) = q_{j+1}(s_{0}), and using the notation δ_{j} = δ_{j}(s_{0}) and ϵ_{j} =

∥u_{t_{j}(s_{0})}(s) − u_{t_{j}(s_{0})}(s_{0})∥, we get \frac{ϵ_{j}}{δ_{j}} − const ϵ_{j}^{2} ≤ ϵ_{j+1} ≤ \frac{ϵ_{j}}{δ_{j}} + const ϵ_{j}^{2}, which is equivalent

to δ_{j}ϵ_{j+1}/(1 + const δ_{j}ϵ_{j}) ≤ ϵ_{j} ≤ δ_{j}ϵ_{j+1}/(1 − const δ_{j}ϵ_{j}). Applying these inequalities recursively, it follows that

const ∥p(s) − p(s_{0})∥ ∏_{i=j}^{m} \frac{δ_{i}}{1 + const δ_{i}ϵ_{i}} ≤ ϵ_{j} ≤ const ∥p(s) − p(s_{0})∥ ∏_{i=j}^{n} \frac{δ_{i}}{1 − const δ_{i}ϵ_{i}} .

It follows from the last sentence in the Sublemma that cϵ_{j} ≤ ∥q_{j}(s) − q_{j}(s_{0})∥ ≤ Cϵ_{j}. Combining this with the above estimates for ϵ_{j}, we get

const ∥p(s)−p(s_{0})∥ ∏_{i=j}^{n} \frac{δ_{i}}{1 + const δ_{i}ϵ_{i}} ≤ ∥q_{j}(s)−q_{j}(s_{0})∥ ≤ const ∥p(s)−p(s_{0})∥ ∏_{i=j}^{n} \frac{δ_{i}}{1 − const δ_{i}ϵ_{i}} .

Dividing all sides of the last row by s − s_{0} and letting s → s_{0}, proves the lemma.

References

[Ba] V. Baladi, Periodic orbits and dynamical spectra, Ergod. Th. & Dynam. Sys. 18 (1998), 255-292.

[B1] R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math. 94 (1972), 1-30.

[B2] R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math. 95 (1973), 429-460.

[BSC] L. Bunimovich, Ya. Sinai,and N. Chernov, Statistical properties of two-dimensional hyperbolic billiards, Russ. Math. Surveys 46 (1991), 47-106.

[Ch1] N. Chernov, Markov approximations and decay of correlations for Anosov flows, Ann. of Math. 147 (1998), 269-324.

[Ch2] N. Chernov, Decay of correlations and dispersing billiards, J. Stat. Phys. 94 (1999),

[D] D. Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math. 147 (1998), 357-390.

[DP] D. Dolgopyat and M. Pollicott, Addendum to ”Periodic orbits and dynamical spectra”, Ergod. Th. & Dynam. Sys. 18 (1998), 293-301.

[I1] M. Ikawa, Decay of solutions of the wave equation in the exterior of several strictly convex bodies, Ann. Inst. Fourier, 38 (1988), 113-146.

[I2] M. Ikawa, Singular perturbations of symbolic flows and poles of the zeta functions, Osaka J. Math. 27 (1990), 281-300; Addendum: Osaka J. Math. 29 (1992), 161-174.

[KB] A. Katok and Burns, Infinitesimal Lyapunov functions, invariant cone families and stochastic properties of smooth dynamical systems, Ergod. Th. & Dynam. Sys. 14 (1994), 757-785.

[KH] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge Univ. Press, Cambridge 1995.

[K] R. Kenny, Estimates of the Hausdorff dimension of the non-wandering set of open billiards, Preprint 1999.

[LP] P. Lax, R. Phillips, Scattering Theory, Revised Ed., Academic Press, London, 1989.

[L] C. Liverani, Decay of correlations, Ann. of Math. 142 (1995), 239-301.

[Mal] B. Malgrange, Ideals of Differentiable Functions, Oxford Univ. Press, London, 1966.

[Mor] T. Morita, The symbolic representation of billiards without boundary condition, Trans. Amer. Math. Soc. 325 (1991), 819-828.

[PP1] W. Parry and M. Pollicott, An analogue of the prime number theorem and closed orbits of Axiom A flows, Ann. Math. 118 (1983), 573-591.

[PP2] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187-188, 1990.

[P] V. Petkov, Analytic singularities of the dynamical zeta function, Nonlinearlity 12 (1999), 1663-1681.

[Po] M. Pollicott, On the rate of mixing of Axiom A flows, Invent. Math. 81 (1985), 413-426.

[PoS] M. Pollicott and R. Sharp, Exponential error terms for growth functions of negatively curved surfaces, Amer. J. Math. 120 (1998), 1019-1042.

[Ra] M. Ratner, Markov partitions for Anosov flows on n-dimensional manifolds, Israel J. Math. 15 (1973), 92-114.

[Ro] C. Robinson, Structural stability of C^{1} flows, Lect. Notes in Math. 468 (1975), 262-277

[R1] D. Ruelle, Thermodynamic formalism, Addison-Wesley, Reading, Mass., 1978.

[R2] D. Ruelle, Resonances for Axiom A flows, Commun. Math. Phys. 125 (1989), 239-262.

[R3] D. Ruelle, The thermodynamic formalism for expanding maps, Commun. Math. Phys. 125 (1989), 239-262.

[Si1] Ya. Sinai, Dynamical systems with elastic reflections, Russian Math. Surveys 25 (1970), 137-190.

[Si2] Ya. Sinai, Development of Krylov’s ideas, An addendum to: N.S.Krylov ”Works on the foundations of statistical physics”, Princeton Univ. Press, Princeton 1979, pp. 239-281.

[Si3] Ya. Sinai, Gibbs measures in ergodic theory, Russ. Math. Surveys 27 (1972), 21-69.

[Sj] J. Sjöstrand, Geometric bounds on the density of resonances for semiclassical problems, Duke Math. J. 60 (1990), 1-57.

[S] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817.

[St1] L. Stoyanov, Exponential instability and entropy for a class of dispersing billiards, Ergod. Th. & Dynam. Sys. 19 (1999), 201-226.

[St2] L. Stoyanov, Asymptotic of travelling times in scattering by two convex bodies, Preprint, 1998.

[WW] J. H. Wells and L.R. Williams, Embeddings and Extensions in Analysis, Berlin, Springer, 1975.

[Y] L.-S. Yang, Statistical properties of systems with some hyperbolicity including certain billiards, Ann. Math. 147 (1998), 585-650.

[Z] M. Zworski, Counting scattering poles, In: Spectral and Scattering Theory, M. Ikawa Ed., Marcel Dekker, New York 1992 ; pp. 301-331.

University of Western Australia, Nedlands WA 6023, Australia E-mail address: stoyanov@maths.uwa.edu.au