ATTRACTORS AND RECURRENCE FOR

DENDRITE-CRITICAL POLYNOMIALS

Alexander Blokh and Michał Misiurewicz

August 23, 2004

Abstract. We call a rational map f dendrite-critical if all its recurrent critical points either belong to an invariant dendrite D or have minimal limit sets. We prove that if f is a dendrite-critical polynomial then for any conformal measure μ either for almost every point its limit set coincides with the Julia set of f, or for almost every point its limit set coincides with the limit set of a critical point c of f. Moreover, if μ is non-atomic then c can be chosen to be recurrent. A corollary is that for a dendrite-critical polynomial and a non-atomic conformal measure the limit set of almost every point contains a critical point.

1. Introduction

The central question in the Dynamical Systems Theory is that of the long term behavior of orbits. To address this question, one often studies ω-limit sets for orbits typical in some sense. In this paper we do it for a class of complex polynomials, understanding “typical” in terms of conformal measures. The paper continues our previous paper on a similar topic ([BM2]) and aims at describing ω-limit sets of points which are realized on sets of positive conformal measure μ. However, in addition to this question, which essentially dates back to Milnor [Mi], we are also interested in the following related problem: is it true that for μ-a.e. point x the ω-limit set ω(x) contains a critical point?

To fix terminology and notation, recall that for a continuous map T of a compact Hausdorff space X to itself and a point x ∈ X the orbit of x is the sequence (f^{n}(x))_{n=0}^{∞} (we denote it orb(x) and sometimes consider it to be a set rather than a sequence), and the ω-limit set of x is the set of all accumulation points of orb(x). We denote the latter ω(x) and usually call it simply the limit set of x.

Let us describe ideas motivating our research. Milnor in [Mi] introduced the notion of an attractor, and, in particular, primitive attractor. For a given measure

Typeset by AMS-TEX

2000 Mathematics Subject Classification Primary 37F10; Secondary 37E25.

Key words and phrases. Complex dynamics, attractors, conformal measures, postcritical set.

This paper was partially written while both authors were visiting Max-Planck-Institut für Mathematik in Bonn during the activity on Algebraic and Topological Dynamics in 2004. Moreover, the first author was partially supported by NSF grant DMS 0140349 and the second author was partially supported by NSF grant DMS 0139916. It is a pleasure for both authors to acknowledge the support of MPIM and NSF.

μ a primitive attractor is a set A such that μ({x : A = ω(x)}) > 0. In the case of rational maps the natural choice for μ is any conformal measure.

The most thoroughly studied case is that of interval maps (see, e.g., [BL1-BL4, Lu, BM1]). In the case of rational complex maps much less has been done. Indeed, working with these maps is more complicated, because the space then is two dimensional. This allows a map to have much more flexibility in terms of its dynamics, which is not always compensated by nice analytic properties of the map. A list of some known results about limit sets of points for rational maps can be found in [BM2].

We need some terminology. If ℂ̂ is the closed complex plane and f : ℂ̂ → ℂ̂ is a rational map, we denote by C(f) = C the set of its critical points and by P (f) = ⋃_{c∈C} ω(c) its postcritical set. A periodic orbit is called a cycle. For a point x let B(x, r) be the open disk of radius r centered at x. A Jordan disk is a set U, homeomorphic to an open disk with \overline{U} homeomorphic to a closed disk such that U is the interior of \overline{U}; closed Jordan disks are closures of the open ones. We call a set A minimal if the map restricted to this set is minimal (i.e., the orbit of every point of A is dense in A). We say that f is exact (for a measure μ) if for any measurable set B such that B = f^{−n}(f^{n}(B)) for n = 0, 1, 2, …, either B or its complement has measure 0. Clearly, an exact map is ergodic.

Suppose that A is a connected set. Then a component V of f^{−n}(A) is said to be an (n)-pull-back of A, and sets f(V), f^{2}(V), …, f^{n−1}(V) are called pull-backs of A corresponding to V. If f is univalent on all V_{i} = f^{i}(V), i < n, then we say that this pull-back is univalent. Suppose that a connected set B ⊂ A (or B ⊃ A) is given. Then any k-pull-back of B contained in a k-pull-back of A corresponding to V (or containing a k-pull-back of A corresponding to V) is also said to be corresponding to V. We will mostly deal with these notions when A or B are Jordan disks, but the definitions can be given in general.

A measure μ on J(f) is conformal (for f) if for an exponent α > 0 we have μ(f(A)) = ∫_{A} |f′(z)|^{α}dμ whenever f|_{A} is 1-to-1 (by [S] f has at least one conformal measure).

Now, for x ∈ ℂ̂ and n > 0 consider the supremum r_{n}(x) of all r such that B(f^{n}(x), r) can be pulled back to x univalently. Then r_{n}(x) > 0 if all points x, …, f^{n−1}(x) are not critical; otherwise define r_{n}(x) as 0. Denote by V the pullback of B(f^{n}(x), r_{n}) corresponding to x; then there exists a critical point c_{n} belonging to the boundary of f^{m_{n}}(V). We call (c_{n}, m_{n}) a generating pair for r_{n}(x).

If x ∈ J(f) and r_{n}(x) ↛ 0 then x is called (C-)reluctant. The set of all such points is denoted by Rlc(f) (reluctant points are also called conical, see, e.g.

[DMNU] and the set Rlc(f) is also called the radial Julia set of f, see [McM]; in Section 8.3 of [LM] reluctant points are discussed in the context of Kleinian groups). If x ∈ J(f) and r_{n}(x) → 0, the point x is called (C-)persistent. There are trivial cases when a point x is persistent, e.g., if it is precritical, that is, eventually mapped into a critical point, or preparabolic, that is, eventually mapped into a parabolic periodic point (whose orbit by the Fatou Theorem is the limit set of some critical point). Given a persistent point x let us call the sequence of generating pairs (c_{n}, m_{n}) with n = 1, 2, … the standard basic setup for the point x. The set of all persistent points is denoted by Prs(f). By the definition, Prs(f) ⊂ J(f) and Rlc(f) ⊂ J(f). Also, let P_{r}(f) be the union of the limit sets of recurrent critical points of f.

The following theorem has been proven in [BMO].

Theorem 1.1. The following holds for a conformal measure μ of a rational map f.

(1) If μ(Rlc(f)) > 0 then supp(μ) = J(f), μ(Rlc(f)) = 1, μ is non-atomic, the map f is exact, conservative, and ω(x) = J(f) for μ-a.e. point x ∈ ℂ̂.

(2) If μ(Rlc(f)) = 0 then for μ-a.e. point x ∈ J(f) either ω(x) ⊂ P_{r}(f) or x is a precritical or a preparabolic point of f.

The aim of Theorem 1.1 is to deal with conformal measures with no assumptions on rational maps f. Observe, that even though the conclusions of Theorem 1.1(1) are quite strong, the conclusions of Theorem 1.1(2) are weaker. Indeed, in the case when μ(Rlc(f)) = 0 one cannot fully describe possible limit sets of μ-a.e. point of J(f) because the inclusion ω(x) ⊂ P_{r}(f) established in Theorem 1.1(2) for μ-a.e. point x is not enough for such description. In other words, Theorem 1.1 does not imply the description of primitive attractors for the measure μ in the case when μ(Rlc(f)) = 0. However, in the present paper we are intersted exactly in the case when μ(Rlc(f)) = 0, and thus we need extra assumptions.

A reasonable type of assumption which one can make here is that about the topology of the Julia set. In fact, for polynomials with connected locally connected Julia sets the primitive attractors can be described similar to the interval case. To explain this we need the notion of backward stability. It extends the Lyapunov (forward) stability to the backward dynamics of a non-invertible map. Essentially, it was first introduced by Fatou who showed that f : ℂ → ℂ is backward stable at points not belonging to the limit sets of critical points; later on it was studied in [Le], [BLe1] and [BO]. The exact definition is as follows. A continuous map f : X → X of a metric space is said to be backward stable at a point x if for any δ there exists ε such that for any connected set K contained in the ε-ball centered at x, any n ≥ 0 and any component M of f^{−n}(K), diam(M) ≤ δ; it is backward stable if it is backward stable at all points. If X is compact then f is backward stable if and only if for any δ there is ε such that for any continuum K with diam(K) ≤ ε, any n ≥ 0 and any component M of f^{−n}(K), diam(M) ≤ δ.

The following theorem was obtained in [BO] (see also [BLe1]).

Theorem 1.2. A polynomial with a locally connected Julia set is backward stable at any point which is not an attracting or neutral periodic as well as on its entire Julia set.

It is known that backward stability implies the description of primitive attractors given below (see, e.g., [BO], [BM2], or [BLe3] for a related direct argument).

Theorem 1.3. For a polynomial f with a locally connected Julia set J(f) and a conformal measure μ one of the following holds.

(1) For μ-almost every x ∈ J(f), ω(x) = J(f).

(2) For μ-almost every x ∈ J(f), ω(x) = ω(c(x)) for some critical point c(x) depending on x, and either x is an eventual preimage of c(x), or x is a preparabolic point, or c(x) is recurrent.

In particular, if μ is non-atomic then for μ-a.e x the point c(x) can be chosen to be recurrent and the set ω(x) contains a critical point. The main corollary then is that for μ-a.e. point x the set ω(x) contains a critical point regardless of whether case (1) or case (2) of Theorem 1.3 takes place. This corollary in fact serves as one of the justifications of our interest to recurrent critical points.

Let us explain the main idea of the proof. By Theorem 1.1 we need to consider a persistent point x and its orbit. We may assume that the limit set of x is not a parabolic orbit (otherwise the situation is trivial). Then it is easy to see that in a generating pair (c_{n}, m_{n}) for r_{n}(x) the point c_{n} can be chosen to belong to J(f), and hence the backward stability on J(f) applies. Then a segment f^{m_{n}}(x), …, f^{n}(x) of the orbit of x is approximated by the corresponding segment c_{n}, …, f^{n−m_{n}}(c_{n}) of the orbit of c_{n}, and thanks to the backward stability this approximation becomes uniformly better and better as n → ∞. Then a dynamical argument, close to that in [BL1], can be applied, yielding that there exists one critical point c(x) = c which on the one hand is approximated by points of the orb(x), and on the other hand has the limit set coinciding with that of x. This shows that unless ω(x) is a parabolic cycle, c(x) ∈ \overline{orb(x)}. Now, if μ is non-atomic we can ignore a countable set of eventual preimages of all critical and parabolic points thus showing that in this case we have c(x) ∈ ω(x) and completing the proof.

On the conceptual level, the main conclusion of Theorem 1.3 is that if the Julia set of a polynomial is relatively “nice” in the topological sense (it is connected locally connected) then it has relatively nice dynamical properties too (given a conformal non-atomic measure μ, primitive attractors in μ-sense can be described and μ-a.e. point contains a critical point in its limit set).

There is another reason which leads to the same conclusion - if the limit sets of recurrent critical points are minimal (and therefore “dynamically nice”) then the same description of attractors follows. Indeed, by Theorem 1.1 we only need to consider persistent points. If the measure μ is non-atomic then we can ignore precritical and preparabolic points. Hence ω(x) ⊂ P_{r}(f) which implies that in fact ω(x) = ω(c(x)) for some recurrent critical point c(x) with minimal limit set. Hence in this case the same description of primitive attractors takes place, and the same conclusion can be made: a μ-a.e. point has a critical point in its limit set.

The main idea of [BM2] was to show that in order to make similar conclusions one need not know that the entire Julia set is topologically nice; it turned out to be sufficient to know that critical points belong to a topologically “nice” forward invariant set. Also, in the conditions on rational maps suggested in [BM2] we combine the above property and minimality of some critical limit sets. This extends the class of maps for which the desired conclusions can be made and in fact simplifies some arguments. To state results of [BM2] we need the following definition: a graph is a one-dimensional branched manifold. Now, we consider in [BM2] rational maps for which each critical point either belongs to an invariant graph G, or has minimal limit set, or is non-recurrent and has the limit set disjoint from G. We call such maps graph-critical and study their attractors. Let us point out that G above is just a topological graph, and hence it is unknown whether graph-critical polynomials have locally connected Julia sets.

By a continuum we mean a non-degenerate (i.e., consisting of more than one point) connected compact set, and by a wandering continuum K we mean a continuum K such that the images of K under all iterates of the map are pairwise disjoint and the trajectory of K does not converge to a periodic orbit.

Theorem 1.4. For a graph-critical rational map f and a conformal measure μ one of the following holds.

(1) For μ-almost every x ∈ J(f), ω(x) = J(f).

(2) For μ-almost every x ∈ J(f), ω(x) = ω(c(x)) for some critical point c(x) depending on x and either x is precritical, or ω(c(x)) is minimal, or there is a wandering continuum K ⊂ G containing c(x) and a recurrent point, or c(x) is recurrent.

The main conclusions of Theorems 1.3 and 1.4 look almost the same. There is however a significant difference between these two theorems. Speaking of conditions, in Theorem 1.3 we deal with polynomials while in Theorem 1.4 we deal with rational maps; also, in Theorem 1.3 we assume that the entire J(f) is locally connected while in Theorem 1.4 no assumptions are made about J(f). Let us now discuss the difference between the conclusions of the two theorems assuming that μ is nonatomic. The main difference is that in Theorem 1.4 we cannot say, that the point c(x) is recurrent (and therefore we cannot conclude that a μ-typical ω-limit set contains a critical point, at least not using our methods). For instance, it could happen so that c(x) ∉ G is not recurrent while ω(c(x)) ⊂ G is minimal and a positive μ-measure of points have ω(c(x)) as their limit set.

However, even if this case is excluded we may not be able to conclude that a μ-a.e. point contains a critical point in its limit set. Indeed, assume that all critical points belong to G. Then one can show (see [BM2]) that in the case (2) of Theorem 1.4 there is a wandering continuum J ⊂ G containing c(x) and a recurrent point, or a point c(x) is recurrent. If wandering continua in G do exist then the former situation may be realized and we will not be able to make the desired conclusion. However, if there are no wandering continua in G then the only possibility is that c(x) is recurrent and thus ω(x) contains a critical point as desired. Hence, our problem is related to the phenomenon of wandering continua, and if we can somehow show that f|_{G} has no wandering continua, then we will achieve our goal. Observe, that wandering continua do not exist in the case when the Julia set is locally connected (see [BLe2]), but may well exist in general.

The existence of wandering continua results in f|_{G} being not backward stable; this is how the fact that f is not backward stable manifests itself. Indeed, the images of a wandering continuum K under a higher and higher iterates of the map must shrink. Hence if we pick a limit point y of the orbit of K we see that components of far away preimages of any neighborhood of y will have to contain K, a contradiction with backward stability.

Therefore, if wandering continua exist, then if we pull back tiny balls around images of a persistent point until they hit critical points, we have no way of knowing that the size of these pull backs is small. In other words, the appropriate orbit segment of a critical point does not necessarily approximate the corresponding orbit segment of the orbit of x better and better as it did before in the backward stable case, and hence the arguments proving Theorem 1.3 break down. Theorem 1.4 is actually proven by means of a subtler argument which in general does not lend itself to straightforward generalizations and does not imply that we can find a recurrent critical point c(x).

In this paper we introduce a new class of rational functions and study polynomials from that class. Let us introduce necessary terminology. A continuum is said to be a dendrite if it is locally connected and tree-like (contains no simple closed curves). A rational map f is dendrite-critical if there exists a forward invariant dendrite D such that every recurrent critical point of f either belongs to D, or has minimal limit set. We will consider dendrite-critical polynomials (so this assumption on the studied class of maps is stronger than before, when we studied graph-critical rational functions), while we only restrict the behavior of their recurrent critical points (so this dynamical assumption is weaker than before). Observe that under our assumptions wandering continua in D may well exist because as before we make only topological assumptions about D; in particular it is not at all clear if J(f) is locally connected for a dendrite-critical polynomial f.

The aim of the present paper is to show that for dendrite-critical polynomials a version of Theorem 1.4 holds. Namely, we prove the following theorem.

Theorem 5.4. For a dendrite-critical polynomial f and a conformal measure μ one of the following holds.

(1) For μ-almost every x ∈ J(f), ω(x) = J(f).

(2) For μ-almost every x ∈ J(f), ω(x) = ω(c(x)) for some critical point c(x) depending on x, and either x is an eventual preimage of c(x), or x is preparabolic, or c(x) can be chosen to be recurrent.

An important ingredient in the proof of Theorem 5.4 is the following theorem.

Theorem 5.3. Suppose that x is a persistent point of a dendrite-critical polynomial f. Then either x is precritical, or x is preparabolic, or there exists a recurrent critical point c(x) such that ω(x) = ω(c(x)).

The following is the main corollary of Theorem 5.4.

Corollary 5.5. If f is a dendrite-critical polynomial and μ is a non-atomic conformal measure, then for μ-a.e. point x the set ω(x) contains a critical point.

The paper is arranged as follows. In Section 2 we go over a dynamical construction from [BM2] which allows one to make conclusions like the ones of Theorem 1.5 in the general setting of continuous maps on a compact metric space. In Section 3 we introduce the notion of recurrent criticality and extend some results of Mañé for this notion. The main result of Section 3 is Theorem 3.5 which could prove to be important for applications. Section 4 is devoted to a detailed study of pull-backs on dendrites under rational and polynomial maps. Finally, we prove the main results of the paper in Section 5.

2. Basic facts about followed points

In this section we list those results of Section 3 of [BM2] needed in what follows. Throughout the rest of this section T : X → X is a continuous map of a metric compact space X with metric d and C ⊂ X is a finite set. In this not necessarily smooth situation one can still call a periodic point a of period m repelling (topologically) if in some metric d_{1} equivalent to d, for some ε > 0 and any point x ≠ a which is at most ε away from a, we have d_{1}(T^{m}(x), a) > d_{1}(x, a). If we use the fact that some point is repelling, we will assume that our metric is already modified as above.

Now we introduce our Basic Setup. It consists of definitions and notation, and depends on a choice of a point x ∈ X and a set C. We give it a special name since we will have to refer to it several times.

Basic Setup. Suppose that x ∈ X and for every integer i ≥ 0 an integer m_{i} ∈ [0, i] and a point c_{i} ∈ C are chosen. Then we use the following definitions and notation.

(1) For a given c ∈ C with infinite sequence of numbers m_{i} with c_{i} = c, if the sequence does not tend to ∞ then we call this case bounded (for c), otherwise we call the case unbounded (for c).

(2) A pair of points (T^{r}(x), T^{r−m_{i}}(c_{i})), m_{i} ≤ r ≤ i, is called an i-pair.

(3) If c ∈ C_{x} then the set of all accumulation points of the sequence (T^{m_{i}}(x)), where c_{i} = c, will be denoted by L_{c}. Clearly, L_{c} ⊂ \overline{orb(x)}. Moreover, in the unbounded case for c we have L_{c} ⊂ ω(x).

(4) A pair of points (x′, c′) which is the limit for some sequence of i-pairs with i → ∞ and c_{i} = c ∈ C with ω(c) not minimal, is called a limiting pair.

□

The following condition was called basic in [BM2].

Basic Condition. We have d(T^{i}(x), T^{i−m_{i}}(c_{i})) → 0 as i → ∞. □

With the Basic Setup, we will say that x is C-followed if Basic Condition holds and for any limiting pair (x′, c′) we have ω(x′) = ω(c′). The simplest Basic Setup is Standard Basic Setup described in Introduction for a persistent point x of a rational map f. Then for any i we have a generating pair (c_{i}, m_{i}) for r_{i} and by the definition of a persistent point Basic Condition is satisfied.

The main general result of [BM2] is that if a point is C-followed then its limit set coincides with the limit set of one of the points of C. Here we provide a bit more detailed statement than the one literally given in [BM2]; the main technical addition is that we emphasize that the point c whose limit set coincides with that of x can be chosen so that it appears infinitely many times in Basic Setup.

Theorem 2.1. If x is C-followed then ω(x) = ω(c) for some c ∈ C. Moreover, this c can be chosen in such a way that c appears in Basic Setup infinitely many times and if the bounded case holds for it then T^{m}(x) ∈ L_{c} for some m ∈ ℕ, while in the unbounded case L_{c} ⊂ ω(x), each point of L_{c} is recurrent and has limit set coinciding with ω(x).

As was explained in Introduction in the case of a map which is not backward stable Standard Basic Setup does not satisfy conditions necessary for the point to be C-followed because in general for limiting pairs (x′, y′) we do not know if ω(x′) = ω(y′). The idea of [BM2] was to suggest a different Basic Setup which uses topological and dynamical properties of graph maps and thus shows that persistent points are C-followed where C is the set of critical points of f. However, the Basic Setup from [BM2] does not imply that c in Theorem 2.1 can be chosen so that c ∈ ω(x); to some extent the aim of this paper is to develop tools which would allow to make such conclusion in some cases.

3. A version of the results of Mañé relative to pull-backs

In this section a version of the results from [Ma] is obtained. The main ideas are from [Ma], the difference being that instead of considering points not belonging to the limit sets of recurrent critical points we consider all points, but at the same time work only with the pull-backs of their neighborhoods which do not contain recurrent critical points. The proofs are close to those by Mañé, but for the sake of completeness and also because the conditions are different we include full proofs in this section (except Lemma 3.1). To state the results we need to introduce the notation which mimics that of Mañé. Let us point out that even though the main results of the paper deal with polynomials, in this section we work with a rational map f.

By a square we understand a square whose boundary segments are vertical and horizontal. The half-length of its side is called the radius of the square, and the point of intersection of its diagonals is said to be its center. Given a square of radius δ centered at p, denote by S^{k} the square of radius kδ centered at p. Also, suppose that U is a Jordan disk and V is a pull-back of U, i.e. V is a component of f^{−n}(U) for some n. Then the number of critical points hit by the sets V, f(V), …, f^{n−1}(V) is said to be the criticality of f^{n}|_{V} (or the criticality of V if n is fixed). On the other hand, the number of recurrent critical points hit by the sets V, f(V), …, f^{n−1}(V) is said to be the recurrent criticality of f^{n}|_{V} (or the recurrent criticality of V if n is fixed). Finally, V is said to be a non-recurrent pull-back of U if f(V), …, f^{n}(V) = U do not contain recurrent critical points.

Note that in this section when a pair of squares with the same center appears, sometimes they are denoted by S and S^{k} with 0 < k < 1, and sometimes by S^{l} and S with l > 1, depending on whether the larger or the smaller square was the original one.

We arrange the section as follows: first we establish a sequence of technical but useful lemmas and then prove the main result. First we include (without proof) Lemma 3.1 proven in [Ma] (see also [P]).

Lemma 3.1. Suppose that numbers ε > 0, 0 < k < 1, γ > 0 and N are given. Then there exists δ = δ(ε, k, γ, N) such that the following holds. Let S be a square of radius less than δ such that d(S, p) > γ for any parabolic or attracting periodic point p. Suppose that V is an n-pull-back of S such that f^{n}|_{V} has criticality at most N. Then any n-pull-back of S^{k} corresponding to V has diameter at most ε.

Essentially, the aim of this section is to prove that in Lemma 3.1 one can replace the assumption about the criticality of V by the corresponding assumption about the recurrent criticality of V. To do so we establish some other technical lemmas and introduce certain constants. First of all, let us denote by ξ a positive number such that any non-recurrent critical point c of f never comes closer than ξ to itself. This implies the following lemma.

Lemma 3.2. Let A be a set such that A, f(A), …, f^{n−1}(A) are sets of diameter less than ξ. Then every non-recurrent critical point of f is covered by at most one set f^{i}(A), 0 ≤ i ≤ n − 1. Also, if A is a pull-back of a Jordan disk f^{n}(A) such that recurrent criticality of f^{n}|_{A} is r then the criticality of f^{n}|_{A} is at most d + r.

Proof. Assume that c is a non-recurrent critical point such that c ∈ f^{i}(A) ∩ f^{j}(A) and i < j. Then f^{j−i}(c) ∈ f^{j}(A) and so we have d(f^{j−i}(c), c) < ξ which contradicts the choice of ξ. The second part of the lemma follows immediately.

The next lemma is similar to Lemma 3.1, but is sometimes more convenient for our purposes.

Lemma 3.3. Suppose that numbers ε > 0, 0 < k < 1, γ > 0 and r are given. Let δ = δ(ε, k, γ, r + d) be the number found in Lemma 3.1. Let S be a square of radius less than δ such that d(S, p) > γ for any parabolic or attracting periodic point p, let V be an n-pull-back of S and V′ be a corresponding to V n-pull-back of S^{k}. Suppose that diam(V′) ≥ ε and the recurrent criticality of f^{n}|_{V} is at most r. Then there exists i, 0 ≤ i ≤ n − 1 such that diam(f^{i}(V)) ≥ ξ.

Proof. Let us assume that the number i with required properties does not exist. Then for any i, 0 ≤ i ≤ n − 1 we have diam(f^{i}(V)) < ξ. This implies by Lemma 3.2 that the criticality of f^{n}|_{V} is at most d + r. Then by Lemma 3.1 and by the choice of δ it follows that we must have diam(V′) < ε, a contradiction with the assumption. Hence there exists i, 0 ≤ i ≤ n − 1 such that diam(f^{i}(V)) ≥ ξ.

Given a square S and a square S^{l} with l > 1 call any square S′ contained in S^{l}\S and having a side intersecting with S and a side intersecting with S^{l} a collar square of S, S^{l} (references to one or even both squares S, S^{l} may be omitted if this does not

cause any confusion). There is a well defined function η(l) = \frac{4l}{l − 1} > 1 such that

the entire “collar” S^{l} \ S is covered by η(l) collar squares. Lemma 3.4 shows that under the assumptions of bounded recurrent criticality the fact that a pull-back of a square S is big implies that a certain collar square to S has a relatively big pull-back too. As we shall see later, together with Lemma 3.3 this implies that yet another square, though smaller than S, has a uniformly bounded away from 0 diameter of one of its pull backs which eventually leads to a contradiction. Of the lemmas proven so far Lemma 3.4 seems to be the most important.

Lemma 3.4. Let numbers l > 1, ε < \frac{ξ}{2}, γ, r be given. Then there exists δ′ = δ′(ε, l, γ, r) such that the following holds. Let S be a square of radius less than δ′ such that d(S^{l}, p) > γ for any parabolic or attracting periodic point p and let V be a n-pull-back of S^{l} with recurrent criticality r. Suppose that there exists an n-pull-back of S corresponding to V and such that its diameter is greater than ε. Then there exists a collar square of S, S^{l} with a pull-back corresponding to V of diameter

at least \frac{ε}{2d^{2d+r}(η(l) + 1)} = ε′.

Proof. As δ′ we choose the number δ′ = δ(ε′, \frac{1}{l}, γ, 2d + r) defined in Lemma 3.1.

Thus, if the distance of a square S̃ from parabolic and attracting points is at least γ and its k-pull-back Ṽ is such that the criticality of f^{k}|Ṽ is at most 2d + r then all pull-backs of S̃^{\frac{1}{l}} corresponding to V have diameters less than ε′ (all that follows from Lemma 3.1).

Suppose that all collar squares of S, S^{l} are such that all their pull-backs corresponding to V have diameters less than ε′. Choose η(l) collar squares of S, S^{l}. Consider for every i ≥ 0 the set f^{−n+i}(S) ∩ f^{i}(V), that is the full preimage of S inside the appropriate pull-back of S^{l}. By the assumption V contains a pull-back of S with diameter greater than ε, so f^{−n}(S) ∩ V has diameter greater than ε. Thus we can find the greatest i for which diam(f^{−n+i}(S) ∩ f^{i}(V)) ≥ ε. Then for all j such that n ≥ j > i we have diam(f^{−n+j}(S) ∩ f^{j}(V)) < ε.

Let us show that then diam(f^{j}(V)) < ε + 2ε′ for each j such that i < j ≤ n. For any point x ∈ f^{j}(V) choose a point x′ ∈ f^{−n+j}(S) ∩ f^{j}(V) so that x, x′ belong to the same pull-back of a collar square of S, S^{l} (x′ will have to be chosen on the boundary of such pull-back). Let us explain how we choose x′ in more detail. If x ∈ f^{−n+j}(S) ∩ f^{j}(V), then we choose x′ = x. If not, then x is the f^{n−j}-preimage of a point from a collar square. Choose the corresponding pull-back of this collar square and a point on the intersection of the boundary of this collar square with S which we also pull back to some point x′.

Since by the construction x, x′ belong to the same pull-back of a collar square, the distance between x, x′ is less than ε′. In other words, any point x ∈ f^{j}(V) can be approximated by a point x′ ∈ f^{−n+j}(S)∩f^{j}(V) such that the distance between x, x′ is less than ε′. On the other hand, the diameter of f^{−n+j}(S) ∩ f^{j}(V) is less than ε by the assumption. The triangle inequality implies that then diam(f^{j}(V)) < ε+2ε′ for each j such that i < j ≤ n. Since ε+2ε′ < 2ε < ξ, we see that diam(f^{j}(V)) < ξ for any j = i + 1, …, n − 1.

Hence by Lemma 3.2 the criticality of f^{n−i−1}|_{f^{i+1}(V)} is at most d + r, and so the

criticality of f^{n−i}|_{f^{i}(V)} is at most 2d + r. Since diam(S^{l}) < δ′ = δ(ε′, \frac{1}{l}, γ, 2d + r)

we conclude that every (n − i)-pull-back of S corresponding to f^{i}(V) has diameter less than ε′. Now, since the criticality of f^{n−i}|_{f^{i}(V)} is at most 2d + r, we see that a collar square of S, S^{l} has at most M = d^{2d+r} (n − i)-pull-backs corresponding to f^{i}(V). On the other hand, we can cover S^{l}\S with η(l) collar squares. Hence, there are no more than Mη(l) of (n−i)-pull-backs of collar squares of S, S^{l} corresponding to f^{i}(V) (and hence contained in f^{i}(V)).

Thus, the set f^{i}(V) is the union of no more than M pull-backs of S, each of which has diameter less than ε′ by the choice of δ′, and Mη(l) pull-backs of the originally chosen η(l) collar squares with each pull-back being of diameter less than ε′ by the assumption. Then the diameter of their connected union V is less than or equal to the sum of all these diameters which is at most Mε′ +η(l)Mε′ = M(η(l)+1)ε′ < ε. However the assumption is that the full preimage of S inside f^{i}(V) - and therefore f^{i}(V) as a whole - have diameter at least ε, a contradiction.

We are ready now to prove the main result of this section. As one can see, Theorem 3.5 almost literally repeats Lemma 3.1 with one main exception: criticality is replaced by recurrent criticality.

Theorem 3.5. Suppose that ε > 0, 0 < k < 1, γ > 0, r are given. Then there exists β = β(ε, k, γ, r) such that the following holds. Let S be a square of radius less than β such that d(S, p) > γ for any parabolic or attracting periodic point p. Suppose that V is an n-pull-back of S such that f^{n}|_{V} has recurrent criticality at most r. Then any n-pull-back W of S^{k} corresponding to V has diameter at most ε and is such that the criticality of f^{n}|_{W} is at most d + r.

Proof. Clearly we may assume that ε < ξ/10. Suppose that for some square S, whose distance from parabolic and attracting periodic points is at least γ, there exists an n_{0}-pull-back U_{0} of S^{k} of diameter less than ε, while the recurrent criticality of the corresponding to U_{0} n_{0}-pull-back of S is at most r. Denote by V the n_{0}- pull-back of S corresponding to U_{0}. Our aim is to construct a sequence of squares S_{0}, S_{1}, … which will have bounded away from 0 diameters of their appropriately chosen pull-backs under bounded from above by n_{0} iterates of f. On the other hand the squares S_{0}, S_{1}, … will have specific sizes converging to 0 and simultaneously guaranteeing that the squares will stay inside a fixed square Ŝ concentric with S and such that S^{k} ⊂ Ŝ ⊂ S. The latter tells us that the squares S_{0}, S_{1}, … will all be no closer than γ to parabolic/attracting points and also that these squares will be contained well inside S on which the n_{0}-pull-back corresponding to U_{0} has recurrent criticality at most r. Therefore, technical lemmas proven above will be applicable provided the original square S is chosen to be small enough. Clearly, this picture eventually leads to a contradiction.

The actual construction relies upon the choice of a convenient parameter l > 1

such that 1 < l < \frac{2k + 2}{3k + 1} (observe that 0 < k < 1 implies that 1 < \frac{2k + 2}{3k + 1} < 2 and

so such number l exists and is always less than 2). For the sake of computations made in this paragraph only we assume that the radius of S is 1 and set S_{0} = S^{k}, so the radius of S_{0} is k. Then the construction will be such that on each step the square S_{i+1} will be non-disjoint from S_{i} and will have the radius equal l − 1 times the radius of S_{i}. Hence the radius of S_{j} is k(l − 1)^{j} and the radii of squares S_{j} form a decreasing geometric progression (recall that l < 2). To estimate how far from the center of S these squares can reach we need to sum up the series

k(1 + 2 ∑_{j=1}^{∞}(l − 1)^{j}) = \frac{lk}{2 − l}. It is easy to see that our choice of l guarantees that

\frac{lk}{2 − l} < \frac{k + 1}{2} = t < 1. Therefore, any sequence of squares described above will

stay inside the square Ŝ = S^{t} and so for any j the distance between S_{j} and any parabolic or attracting point is less than γ, which makes previously proven lemmas applicable to squares S_{j} provided the size of the original square S is appropriately small. Observe that if S_{j}, j ≥ 1, is one of our squares, then the square S_{j}^{2} is contained in S.

Now that the number l has been chosen we can choose β. To do so set l′ = \frac{l + 1}{2} and then apply Lemma 3.4 and choose the number δ′ = δ′(ε, l′, γ, r). Thus, if there is a square R such that diam(R) ≤ δ′, R^{l′} ⊂ S and there is a pull-back of R corresponding to V and of diameter greater than ε then there is a collar square of R, R^{l′} whose appropriate pull-back is of diameter at least \frac{ε}{2d^{2d+r}(η(l′) + 1)} =

ε′. Then we apply Lemma 3.1 and find the number δ = δ(ε′, 1/2, γ, r + d). By Lemma 3.3, δ has the following property: if a square R ⊂ S is such that R^{2} ⊂ S and also R has a pull-back corresponding to V and such that the diameter of this pull-back is at least ε′ then there exists a pull-back of R^{2} corresponding to V and of diameter greater than ξ. Now, as our number β we choose the smaller of δ, δ′; this ensures that both Lemmas 3.3 and 3.4 will be applicable throughout the argument.

Let us now prove that β has the properties from the lemma. Assume that this fails. Then for some square S of radius less than β, whose distance from parabolic and attracting periodic points is at least γ, there exists an n_{0}-pull-back U_{0} of S^{k} of diameter at least ε while the recurrent criticality of the corresponding to U_{0} n_{0}-pull-back of S is at most r. Denote by V the n_{0}-pull-back of S corresponding to U_{0}. To show that this leads to a contradiction we construct a sequence of squares following the ideas described above.

The first square in the sequence is S^{k} = S_{0}. To choose S_{1} we proceed as follows. Our assumption is that U_{0} is an n_{0}-pull-back of S_{0} of diameter at least ε corresponding to an n_{0}-pull-back V of S such that the recurrent criticality of f^{n_{0}}|V is at most r. Then by Lemma 3.4 and by the choice of β we can choose a collar square S′ of S_{0}, S_{0}^{l′} such that there exists a pull-back V′ of S′ corresponding to V with diam(V′) ≥ \frac{ε}{2d^{2d+r}(η(l′) + 1)} = ε′. Because of the choice of β we then can apply

Lemma 3.3 to the square S′^{2} = S_{1} ⊂ S and conclude that it has a pull-back U_{1} corresponding to V which has diameter at least ξ > ε. Observe that the radius of the square S_{1} is l − 1 times the radius of the square S_{0} (this is exactly why we needed one more constant l′). We can assume that U_{1} is an n_{1}-pull-back of S_{1} corresponding to V and then n_{1} ≤ n_{0}. By the arguments from the second paragraph of this proof, S_{1} ⊂ S^{t} ⊂ S and so the distance between S_{1} and any parabolic or attracting point of f is at least γ and we can repeat the argument.

Literally the same arguments apply to the square S_{i} which will be constructed after i steps in the process described above. That is, by the construction we will know that S_{i} has an n_{i}-pull-back of diameter at least ε; moreover, all squares S_{j} are consecutively non-disjoint and of radii k(l − 1)^{j} respectively, and n_{0} ≥ n_{1} ≥ ⋯ ≥ n_{i}. Then we will apply Lemma 3.4 and find a collar square R of S_{i}, S_{i}^{l′} whose appropriate n_{i}-pull-back is of diameter greater than ε′. Then we will apply Lemma 3.3 and show that the square R^{2} has an appropriate n_{i+1}-pull-back of diameter a least ξ, and hence of diameter greater than ε. Moreover, we will have n_{i+1} ≤ n_{i}. Hence the construction can be repeated infinitely many times. However, then the radius of S_{j} converges to 0 and the iterates n_{j} of f for the pull-backs U_{j} of diameter greater than ε stay less than n_{0}, which is clearly impossible. This contradiction proves the first part of the claim of the theorem; the second part dealing with criticality follows immediately from the first one and Lemma 3.2.

As an application of Theorem 3.5 consider a point x which does not belong to the union P_{r} of the limit sets of recurrent critical points of f and to the set of parabolic and attracting periodic points of f. Then if δ is small enough, the square S of radius δ centered at x is disjoint from P_{r} and is distance from the set of parabolic and attracting periodic points of f is at least γ. Hence recurrent criticality of any pull-back of S is zero and moreover Theorem 3.5 applies. By this theorem, for given 0 < k < 1 we can find β such that any square S^{k} has only pull-backs of diameter at most ε, and so at the point x the map f is backward stable. Thus, we obtain another proof of one of the results of [Ma].

4. Dendrites

This section is devoted to studying pull-backs of dendrites under rational and polynomial maps. The main problem with using dynamical properties of maps of dendrites in our circumstances is that because we consider all points on the plane we need to consider their orbits and pull-backs under the plane map in question and not just under the restriction of this map onto the dendrite. Therefore, if U is an open connected set such that U ∩ D is connected (here D is a dendrite), then a decent pull-back of U in the sense of a rational map may have a disconnected intersection with the dendrite and thus may well correspond to two or more pull-backs of U ∩D taken in the 1-dimensional sense. Clearly, studying of possible cases when this phenomenon takes place is rather important for us, and if we can exclude it one way or another this would allow us to proceed with the tools developed in [BM2]. In this investigation we will need some easy topological properties of dendrites; in particular, it is well-known (see, e.g., [BO]) that dendrites are uniquely arcwise connected, which will be used later on.

Some tools which allow us to fight this problem were suggested in [BM2]. More precisely, what was done in this direction in [BM2] is that in the case of a forward invariant graph G one can extend it and construct a larger forward invariant graph G′ such that for G′ the breakdown of connectivity described in the preceding paragraph can only happen after at least one critical point was hit by the pull-backs of U. This was enough in [BM2] because we only needed to pull back neighborhoods until a critical point is hit for the first time. However this is not enough in the present paper, so another set of tools is needed.

Lemma 4.1. The image of a Jordan curve contained in ℂ under a polynomial cannot be simply connected.

Proof. Let f be a polynomial, let H be a Jordan curve bounding a set G, and suppose that f(H) is simply connected. Then the boundary of f(G) is contained in f(H), but f(G) is bounded and has a nonempty interior, a contradiction.

Lemma 4.2. Let f be a polynomial. Suppose that U ∈ ℂ is a closed Jordan disk and D is a dendrite such that D ∩ U is connected. If the boundary of U contains no critical values of f and V is a 1-pull-back of U such that D ∩ U contains images of all critical points of f|_{V}, then V is a closed Jordan disk and f^{−1}(D) ∩ V is a dendrite.

Proof. We can cover the intersection of D and the boundary of U by finitely many small Jordan disks U_{1}, …, U_{k} and find a subdendrite D′ ⊂ D, contained in the interior of U, and intersecting each U_{i} ∩ D. Since the boundary of U contains no critical values of f, if U_{i}’s are sufficiently small, then each component of f^{−1}(U_{i}) is mapped by f homeomorphically onto U_{i}. Therefore, if we prove that f^{−1}(D′) ∩ V is connected, it will follow that f^{−1}(D) ∩ V is also connected. Then, since f is a polynomial (so it has nice local structure), the only reason why f^{−1}(D) ∩ V would not be a dendrite could be that it contained a loop. However, this is impossible by Lemma 4.1.

Thus, it remains to prove that f^{−1}(D′) ∩ V is connected and V is a closed Jordan disk. Clearly, we may assume that D′ ∩ U contains images of all critical points of f|_{V}. Therefore U \D′ is topologically an annulus and f maps V \f^{−1}(D′) onto it as a local homeomorphisms. By the Riemann-Hurwitz formula, the set V \ f^{−1}(D′) has Euler characteristic zero, so it is also topologically an annulus. Therefore f^{−1}(D′) ∩ V is connected and V is a closed Jordan disk.

Lemmas 4.1 and 4.2 are useful tools in studying pull-backs of dendrites under rational maps. In the next lemma speaking of distances between points we use the standard spherical metric on S^{2}. By P_{r}(f) we denote the union of limit sets of recurrent critical points of f.

Lemma 4.3. Let f be a dendrite-critical polynomial and D be the dendrite from its definition. Then there exists a forward invariant dendrite D′ ⊂ D with respect to which f is dendrite-critical, and a number ε > 0, for which the following holds: whenever W is a closed Jordan disk W such that diam(W) < ε, the boundary of W is disjoint from the critical orbits of f, W intersects P_{r}(f) and W ∩D′ is connected, then for every pull-back V of W the set V ∩ D′ is also connected.

Proof. Recall that we consider f as a map on the sphere, and so the case when D contains N will have to be considered. In fact, this case is a bit harder to tackle, so to begin with we assume that D does not contain N. We show that then we can set D′ = D.

Suppose that U is a closed Jordan disk such that U does not contain N, the boundary of U is disjoint from critical orbits of f, and U ∩ D is connected. Let us prove that then for any 1-pull-back V of U the set V ∩ D is connected and V is a closed Jordan disk. Choosing as ε the distance between D and N and applying this claim inductively to a closed Jordan disk W with properties from the lemma, we will complete the proof.

Suppose that V ∩ D is not connected. Construct a new dendrite T in such a way that T ⊃ D, T contains all critical values of f and T ∩ U is connected. Then by Lemma 4.2 f^{−1}(T ∩ U) ∩ V is a dendrite and V is a closed Jordan disk. Since by the assumption f^{−1}(D) ∩ V is not connected then there exist at least two distinct components of f^{−1}(D)∩V. We can find an arc I ⊂ f^{−1}(T ∩U)∩V connecting them so that all points of I (except for the endpoints a, b) do not belong to D. Connect a and b with the arc J inside D. Then I ∪ J = H is a Jordan curve. Observe that by the construction and the assumptions H does not contain N while the image of H is contained in T and is therefore simply connected. This contradicts Lemma 4.1 and shows that f^{−1}(D) ∩ V is connected. Observe that here the size of U does not matter.

It remains to consider the case when N ∈ D. Then N is a cut point of D. The components of D \ N are mapped one onto another in a well-defined fashion because N is fully invariant. Choose a closed Jordan disk U containing N in its interior and show that then there are only finitely many components of D \ N not contained in U. Indeed, otherwise we can choose a sequence of such components B_{1}, B_{2}, … so that there are points x_{i} ∈ B_{i} ∩ ∂U with x_{i} → x ∈ ∂U. Then small neighborhoods W of x are such that W ∩ D is disconnected (because it has points of all B_{i} with big i and B_{i} are distinct components of D \ N), a contradiction to the local connectivity of D. Hence for every U there are only finitely many B_{i}’s not contained in U.

Hence there are finitely many components of D \ N intersecting P_{r}(f). Denote these components of D \ N by A_{1}, …, A_{n}. Let us show that the map f permutes them. Indeed, given A_{l} there exists a point x ∈ ω(c)∩A_{l} where c ∈ D is a recurrent critical point. Then there exists a point y ∈ ω(c) such that f(y) = x. Choose r so that y ∈ A_{r}. Then clearly f(A_{r}) ⊂ A_{l}. In other words, f maps A_{i}’s so that the corresponding map φ of the finite set {1, …, n} is surjective. Hence φ is a permutation as desired. Set D′ = N ∪ ⋃_{i=1}^{n} A_{i}. Clearly, f is dendrite-critical with respect to D′. Choose δ > 0 smaller than the distance between N and the filled-in Julia set K(f). Then for every pair of subscripts i, j the distance between the sets A_{i} \ B(N, δ) and A_{j} is positive. Choose ε < δ which is less than the minimal distance between any two such sets.

Assume that W is a closed Jordan disk of diameter less than ε, intersecting a set A_{i} (e.g., this is so if W intersects P_{r}(f)). Observe that then i is unique and W does not contain N. Suppose that R is a k-pull-back of W intersecting D′. Then R ∩ D′ = R ∩ A_{j} where the set A_{j} is a unique set from the collection A_{1}, …, A_{n} with f^{k}(A_{j}) ⊂ A_{i}. Suppose that for a 1-pull-back V of W the intersection V ∩A_{j} is not connected. Create a new dendrite T ⊃ (D′ ∩ W) so that T contains all critical values of f in W and T ∩ W is connected. By Lemma 4.2 the set f^{−1}(T) ∩ V is a dendrite. Since by the assumption V ∩D′ = V ∩A_{j} is not connected, we can find an arc I with the endpoints a ∈ A_{j}, b ∈ A_{j} such that the image of I is contained in T and I is disjoint from D′ except at a, b. Also, connect a, b with the arc J inside A_{j}. Then J is contained in A_{j} and avoids N. The arc I avoids N by the construction. Thus, the Jordan curve I ∪ J avoids N while its image is contained in the dendrite T, a contradiction with Lemma 4.2. This completes the proof of the lemma.

Finally we prove the following lemmas in which the standard interval notation is adopted.

Lemma 4.4. Suppose that f is a rational map with an invariant dendrite G. Let I ⊂ G be a connected set such that I ∩ f(I) ≠ ∅ and

lim inf_{n→∞} diam(f^{n}(I)) = 0.

Then the set K_{0}^{∞} = ⋃_{j=0}^{∞} f^{j}(I) is connected and the following possibilities hold:

(1) the set K_{0}^{∞} contains a neutral fixed point;

(2) the orbit of I converges to an attracting fixed point;

(3) the orbit of I converges to a parabolic fixed point a so that for some point d ∈ G the component U of G\{d, a} containing (d, a) is such that f(U) ⊂ U, all points of U are attracted to a and I is mapped inside U by some iterate of f; moreover, the point d can be chosen arbitrarily close to a.

Thus, if K_{0}^{∞} does not contain a neutral fixed point then lim_{n→∞} diam(f^{n}(I)) = 0.

Proof. Since I intersects f(I), then also f^{n}(I) intersects f^{n+1}(I) for each n. Therefore the sets K_{m}^{n} = ⋃_{j=m}^{n} f^{j}(I) (including the case n = ∞) are connected. Since lim inf_{n→∞} diam(f^{n}(I)) = 0, there is a sequence k_{n} → ∞ with f^{k_{n}}(I) → a, a ∈ G. Then f^{k_{n}+1}(I) → f(a), and since diam(f^{k_{n}}(I) ∪ f^{k_{n}+1}(I)) → 0 we get f(a) = a.

Let us show that a must be attracting or neutral. Indeed, suppose that a is repelling. Then for any point x closer than some δ > 0 to a we have d(f(x), a) > d(x, a). Now, consider two cases. First assume that a ∈ I. To see that

lim inf_{n→∞} diam(f^{n}(I)) = 0

is impossible, choose small ε > 0 so that ε < δ and I is not contained in the ε-ball B centered at a. By connectivity there exists a point y ∈ I such that d(y, a) = ε. Thus d(f(y), a) > ε and f(I) is not contained in B either. By induction it implies that diam(f^{k}(I)) > ε for every k, a contradiction. Now, consider the case when a ∉ I. Choose ε < δ so small that the ε-ball B centered at a is disjoint from I. Let us show by induction that no image of I is contained in B. Indeed, this is true for I. If it fails, it has to fail for the first time for some n. Then f^{n−1}(I) is partially outside B while on the other hand it intersects f^{n}(I) ⊂ B. Hence by connectivity there is a point y ∈ f^{n−1}(I) whose distance from a is ε. This implies that d(f(y), a) > ε while on the other hand f(y) ∈ f^{n}(I) ⊂ B, a contradiction. So in any case there exists a ball B centered at a such that f^{n}(I) ⊄ B for any n. Clearly, it contradicts the fact that f^{k_{n}}(I) → a.

If an image of I contains a neutral fixed point then (1) holds and there is nothing to prove. If a is an attracting fixed point then the fact that f^{k_{n}}(I) → 0 implies that the orbit of I converges to this fixed point as desired and (2) holds. So from now on we may assume that neither (1) nor (2) takes place. That is, a is a neutral fixed point and the set K_{0}^{∞} does not contain a. We may also assume that I is closed.

Since G is a dendrite, the set K_{0}^{∞} is contained in a unique component A of G \ \{a\}. Consider the unique arc J = [x, a) ⊂ \overline{A} such that J ∩ I = \{x\} (recall that we use usual interval notation here). Then J ⊂ K_{0}^{∞} because of the properties of dendrites. Each point z ∈ J defines an open connected set U_{z} ⊂ G which is a component of G \ {z, a} containing (z, a). Since a is neutral we may assume that the point z is chosen so that \overline{U_{z}} contains no critical points of f and the only fixed point of f in \overline{U_{z}} is a. Since J is contained in K_{0}^{∞}, we have f(z) ∈ K_{0}^{∞} ⊂ A. Consider two cases.

(i) f(z) ∉ \overline{U_{z}}. Then the choice of z guarantees that f([z, a]) ⊃ [z, a] and that points of J which are mapped by f back into J, are mapped farther away from a. Then the argument mimicking the previous argument dealing with repelling periodic points shows that in this case we have a contradiction. Indeed, take the first m such that f^{m}(I) ⊂ U_{z}. Since f^{m−1}(I) intersects f^{m}(I), we see that f^{m−1}(I) intersects U_{z}. By the choice of m we see that f^{m−1}(I) is not contained in U_{z}. Therefore by connectivity z ∈ f^{m−1}(I) and so f(z) ∈ f^{m}(I) ⊂ U_{z}, a contradiction.

(ii) f(z) ∈ \overline{U_{z}}. By the choice of z we know that f(z) ≠ z. Then f(z) ∈ U_{z}. Consider f([z, a])∩[z, a]; clearly, this is an interval of the form [b, a] where b ∈ [z, a] (we rely upon the fact that a is an endpoint of \overline{A}). Consider the point d ∈ [z, a] such that f(d) = b. Then there are two possibilities. First, d may belong to (b, a). In this case the situation is like the one in (i), which leads to the contradiction.

Second, d may belong to [z, b). In this case points on the interval [d, a] are pushed closer to a on the same interval. We will show that then f(U_{d}) ⊂ U_{b}. Indeed, the map on \overline{U_{d}} is injective. Therefore, the only point of \overline{U_{d}} mapped by f to b is d. Given any other point u ∈ U_{d}, connect it by the unique arc [u, v] to [d, a] (so that v ∈ (d, a)). Then the image of [u, v] is the arc connecting f(v) ∈ (b, a) and f(u). If f(u) ∉ U_{b} then there exists a point t ∈ [u, v] mapped to b or to a, a contradiction with injectivity of f|\overline{U_{d}}. This actually implies that the orbits of all points of U_{d} converge to a. Indeed, by the above argument f(U_{d}) ⊂ U_{b}, where b = f(d). This can be repeated, which by induction implies that f^{n}(U_{d}) ⊂ U_{f^{n}(d)}. Since there are no other fixed points in [z, a), we conclude that U_{b} can serve as the set U from the case (3) of the lemma.

Observe that if neither (1) or (2) holds then Snail Lemma easily implies that a is parabolic.

An easy analysis of the result proven in Lemma 4.4 using information about rational maps and their parabolic points leads to the following corollary given here without proof. To state it denote by A the component of G \ {a} containing U.

Corollary 4.5. In the situation of Lemma 4.4(3) the point d can be chosen in such a way that \overline{U} ∩ J(f) = \{a\}. Moreover, the distance between U and the set A ∩ J(f) is then positive.

We are now ready to prove Lemma 4.6 which will be applied in the next section.

Lemma 4.6. Suppose that f is a rational map with an invariant dendrite D. Suppose that [a_{n}, b_{n}] = I_{n}, i = 1, 2, …, is a sequence of arcs in D converging to an arc I′ = [a, b] and such that a_{n}, b_{n} ∈ J(f). Moreover, suppose that there exists a sequence (m_{n})_{n=1}^{∞} such that diam(f^{m_{n}}(I_{n})) → 0 and f^{m_{n}}(I_{n}) does not contain neutral periodic points for any n. Then I = (a, b) is wandering, and so diam(f^{k}(I′)) → 0 and ω(a) = ω(b).

Proof. We may assume that a ≠ b (therefore the images of I are never degenerate) and m_{n} → ∞. Observe that a, b ∈ J(f). Assume that I = (a, b) is not wandering. Then there exist two positive integers k, l such that f^{k}(I) ∩ f^{k+l}(I) ≠ ∅, and so there are points x, y ∈ I such that f^{k}(x) = f^{k+l}(y). For the sake of simplicity we assume that k = 0 and l = 1; the same arguments can be repeated in general with the appropriate changes (e.g., one will have to consider a certain iterate of f and not f itself, etc.).

By the assumptions, every compact subinterval K of I is contained in all but finitely many I_{n}’s and if K is big enough then it contains both x and y so that K ∩ f(K) ≠ ∅. Hence Lemma 4.4 and Corollary 4.5 apply to K. Observe that the images of K cannot contain neutral periodic points because of the assumptions on I_{n}. Thus we need to consider cases covered by Lemma 4.2 (2) and (3).

First consider a simpler case, covered in Lemma 4.4(2), when the orbit of K converges to an attracting fixed point. Each I_{n} contains points of J(f), and so does f^{m_{n}}(I_{n}). On the other hand, if n is big then f^{m_{n}}(I_{n}) ⊃ f^{m_{n}}(K) and the set f^{m_{n}}(K) gets closer and closer to the attracting fixed point a as n grows to infinity. Thus, if ε > 0 is less than one half of the distance between a and J(f) then from some time on diam(f^{m_{n}}(I_{n})) > ε, a contradiction.

Suppose now that the parabolic case covered in Lemma 4.4(3) takes place. Fix a number N such that f^{N}(K) ⊂ U where U is chosen as in Lemma 4.4(3) and Corollary 4.5. Clearly, for all big n we may assume that m_{n} > N and I_{n} ⊃ K. Hence f^{m_{n}}(I_{n}) is a set which on the one hand contains points of U and on the other hand contains points of J(f). Moreover, the fact that f^{m_{n}}(I_{n}) does not contain neutral periodic points implies that f^{m_{n}}(I_{n}) is contained in the same component of D \ {a} as U. Therefore by Corollary 4.5 we see that diam(f^{m_{n}}(I_{n})) > ε for some ε > 0 and all sufficiently large n, a contradiction.

5. Main results

The main aim of this section is to prove Theorem 5.3 as well as its corollaries. First let us introduce some terminology. Recall that P_{r}(f) is the union of limit sets of recurrent critical points of f. Also, denote by P A(f) the union of all parabolic periodic points of f.

The next lemma is useful in the proof of Theorem 5.3; it is used in the proof of Theorem 1.1 and will be useful for us as well.

Lemma 5.1 [BMO]. If z ∈ Prs(f) is neither precritical nor preparabolic then ω(z) ⊂ P_{r}(f).

We will also need the following well-known lemma from topological dynamics (for the proof see, e.g. [BM2]).

Lemma 5.2. Let X be a compact metric space and T : X → X a continuous map. Let x ∈ X, M > 0, and let K ⊂ ω(x) be a compact set such that T^{M}(W ∩ω(x)) ⊂ K for some open set W ⊃ K. Then ω(x) = ⋃_{i=0}^{M−1} T^{i}(K). In particular, if there are pairwise disjoint non-empty compact invariant sets A_{1}, …, A_{j} whose union is ω(x) then j = 1 and A_{1} = ω(x). Moreover, finite limit sets are cycles.

Before passing on to Theorem 5.3, we would like to point out that even though the restrictions included in the definition of a dendrite-critical rational map do not seem to be too strong, still one can show that dendrite-critical rational maps cannot have Cremer points. Indeed, suppose that this is false. Then without loss of generality we may assume that there exists a fixed Cremer point a. It is known that then a belongs to the limit set of a recurrent critical point c [Ma]. By the assumptions on dendrite-critical maps this implies that c ∈ D, and hence a ∈ D. Choose a small neighborhood U of a so that U ∩ D is connected and c ∉ \overline{U}. Local connectedness of D implies that there are only finitely many components of D \{a} which are not contained in U. Denote those of them which contain points of ω(c) in their intersections with U by A_{1}, A_{2}, …, A_{l}. Also, since D is invariant then the following holds: for every component K of D \{a} there exists a unique component L of D \ {a} such that f(K ∩ U) ⊂ L.

Now, let us apply f to A_{1} ∩ U. By the definition it either gets mapped into A_{i} for some A_{i}, or it gets mapped into a small component K ⊂ U of D \ {a}. In the latter case we can apply f over and over until the component in question maps into some A_{i} for the first time. This must happen because c must eventually exit U (after all, c ∉ \overline{U} is recurrent). Hence in either case we will find the next A_{i} to which the same arguments can be applied. Clearly this eventually leads to some j such that a small arc I = [a, b] ⊂ A_{j} ∩ U eventually maps into A_{j}. Choosing an even smaller arc I′ inside A_{j} ∩ U with an endpoint at a we see that this arc will map over itself or into itself under an appropriate finite iterate of f. By Snail Lemma this is impossible for a Cremer point a, a contradiction.

We can now pass on to Theorem 5.3.

Theorem 5.3. Suppose that x is a persistent point of a dendrite-critical polynomial f. Then either x is precritical, or x is preparabolic, or there exists a recurrent critical point c(x) such that ω(x) = ω(c(x)).

Proof. Assume that x is neither preparabolic nor precritical. Then by Lemma 5.1 ω(x) ⊂ P_{r}(f). Now, just like in the definition of a dendrite-critical rational map there are two main requirements on recurrent critical points of f, there are two cases we need to deal with here. The first case is easier so we begin with it. Namely, suppose that ω(x) ⊄ D. Then the set ω(x) \ D is contained in the union of all limit sets of recurrent critical points contained in S^{2} \ D. Consider the union B of all recurrent critical points c ∈ S^{2}\D such that ω(c)∩(ω(x)\D) ≠ ∅. By the definition of a dendrite-critical map, any limit set ω(c), c ∈ B, is minimal, and since all these sets ω(c) intersect ω(x), then B′ := ⋃_{c∈B} ω(c) ⊂ ω(x). On the other hand, since D is invariant and all sets ω(c) with c ∈ B are minimal, B ∩ D = ∅ implies that B′ ∩ D = ∅. Hence, ω(x) = B′ ∪ (D ∩ ω(x)), with both sets in the union invariant and disjoint. Since B′ ≠ ∅, it follows from Lemma 5.2 that ω(x) = B′, and by the same lemma B′ = ω(c) for some recurrent critical point c ∈ B which completes the case when ω(x) ⊄ D.

Let us now assume that ω(x) ⊂ D. Denote by A the union of the set of all critical points of f and the set of all its parabolic periodic points. We want to choose Basic Setup so that x is A-followed. Then by Theorem 2.1 we will have that ω(x) = ω(c(x)) where c(x) ∈ A. Moreover, some extra properties which we will establish guarantee that c(x) ∈ ω(x).

Loosely, the idea is as follows. First, choose for every point f^{n}(x) in the orbit of x a certain neighborhood W_{n} of f^{n}(x) and a certain k_{n}-pull-back V_{n} of W_{n}. The pull backs will be such that V_{n} ∋ c_{n} where c_{n} ∈ A. Then we set m_{n} = n − k_{n} and thus complete Basic Setup - that is, up to the choice of the crucial elements of the construction which are W_{n}, V_{n} and k_{n} and up to the proof that x is A-followed. Now, certain sets on this list are not difficult to come up with. Indeed, we have total control over neighborhoods W_{n}, so they can be chosen in such a way that their diameters converge to 0 (in addition we want them to have connected intersections with D). However, we do not have control over V_{n}, and this is when the tools developed in Sections 3 and 4 become helpful.

We may assume that D has the properties from Lemma 4.3 and for any sufficiently small ε and any Jordan disk W such that W ∩ D is connected and W contains points of the set P_{r}(f), all pull-backs V of W intersect D over a connected set.

Let us pass to the precise construction. It is done in steps, so let us describe the m-th step assuming that 1/m < ε where ε is the constant found in Lemma 4.3. We can also assume that ε is less than the distance between the union of all attracting periodic points of f and the Julia set J(f), and that ε is less that the distance between any limit set of a recurrent critical point c not belonging to D and D itself (recall that such limit sets are minimal and therefore disjoint from D). Set ε_{m} = 1/m and cover D with a finite collection U of closed Jordan disks U_{1}, …, U_{k} whose intersections with D are connected (this is possible since D is locally connected), whose boundaries are disjoint from the orbits of critical points, and whose diameters are less than ε_{m}. Then we choose a Lebesgue number δ′ for this cover. On the other hand we choose the number β = β(ε_{m}, 1/2, γ, 1) from Theorem 3.5, where γ is the minimal distance between any \overline{U_{i}} disjoint from P A(f), and P A(f). By Theorem 3.5 if we have two squares T and T^{2}, the diameter of T^{2} is less than β, d(T^{2}, P A(f)) > γ and there is a pull-back V of T^{2} such that among sets V, f(V), …, T^{2} the only one containing a recurrent critical point is V, then any corresponding to V pull-back of T will have diameter at most ε_{m}. Finally, we set δ_{m} = min(δ′/10, β).

The above implies that given a point y ∈ D, whose distance from P A(f) is at least γ, we can first find j such that the ball of radius δ′ centered at y is contained in U_{j}, and then a square S of radius δ_{m} centered at y. If we consider a pull-back of U_{j} of recurrent criticality 1 then corresponding to it pull-back of S will be of diameter at most ε_{m}. This fact which follows from of Theorem 3.5 plays an important role in the construction below.

Choose N_{m} so big that r_{n}(x) < δ_{m} for every n ≥ N_{m}. The m-th step in the construction will be valid for the numbers n such that N_{m} ≤ n < N_{m+1}. Let us explain how we choose Basic Setup. Given n ≥ N_{m} we first measure the distance between f^{n}(x) and the parabolic periodic points of f. If there exists a parabolic periodic point a such that d(f^{n}(x), a) ≤ ε_{m} then we set c_{n} = a, m_{n} = n. Suppose that the distance between f^{n}(x) and the parabolic periodic points of f is greater than ε_{m}. Then any set U_{j} containing f^{n}(x) has the closure disjoint from P A(f) because the diameter of any U_{j} is less than ε_{m}. Choose a set U_{i(n)} ∋ f^{n}(x) in such a way that the ball of radius δ_{m} centered at f^{n}(x) is contained in U_{i(n)} (it is possible because δ′ is a Lebesgue number of U).

The distance of the set U_{i(n)} from P A(f) is at least γ, so it is disjoint from P A(f). Pull back U_{i(n)} until it hits a recurrent critical point c, or until it hits the critical point c′ generating r_{n}, whichever comes first. Set c_{n} = c or c_{n} = c′ respectively. This defines the number m_{n} and ultimately our Basic Setup.

What needs to be proven now is that x is A-followed and c(x) ∈ ω(x). To prove that x is A-followed we need to verify several properties listed in Section 2. First, observe that Basic Condition is satisfied because ε_{m} → 0 and d(f^{n}(x), f^{n−m_{n}}(c_{n})) < ε_{m} if N_{m} ≤ n < N_{m+1}, so d(f^{n}(x), f^{n−m_{n}}(c_{n})) → 0 as n → ∞. The main part of the verification of the fact that x is A-followed is to check if for any limiting pair (x′, c′) we have ω(x′) = ω(c′). Consider a sequence of n-pairs with n → ∞ which converge to a pair of points (x′, c′). That is, suppose that (x′, c′) = lim(f^{l_{n}}(x), f^{l_{n}−m_{n}}(c_{n})) along a sequence of numbers n and with m_{n} ≤ l_{n} ≤ n. To prove that ω(x′) = ω(c′) we make use of the fact that D is a dendrite.

Suppose that there is a sequence of n-pairs (f^{l_{n}}(x), f^{l_{n}−m_{n}}(c_{n})) converging to (x′, c′) and such that in all of them c_{n} is a critical point generating r_{n}. This means that in the construction the pull-backs of U_{i(n)} do not hit recurrent critical points until they reach c_{n}. Therefore by Theorem 3.5 we have d(f^{l_{n}}(x), f^{l_{n}−m_{n}}(c_{n})) < ε_{k} = 1/k where k is such that N_{k} ≤ n < N_{k+1}. On the other hand, k → ∞ as n → ∞. Therefore in this case d(f^{l_{n}}(x), f^{l_{n}−m_{n}}(c_{n})) → 0 as n → ∞ and so x′ = c′ and ω(c′) = ω(x′) as desired. Observe that in this case we essentially prove that if there is a sequence of n-pairs (f^{l_{n}}(x), f^{l_{n}−m_{n}}(c_{n})) converging to (x′, c′) and such that c_{n} is a critical point generating r_{n} then x′ = c′.

Suppose that there is a sequence of n-pairs (f^{l_{n}}(x), f^{l_{n}−m_{n}}(c_{n})) converging to (x′, c′) and such that in all of them c_{n} is a parabolic periodic point. Then by the construction in this case n = m_{n} and hence l_{n} = m_{n} = n which implies that we have n-pairs (f^{n}(x), c_{n}) converging to (x′, c′). Clearly, this implies that c′ is a parabolic periodic point and that x′ = c′.

¿From now on we may assume that in our sequence of n-pairs giving rise to the limiting pair (x′, c′) all n-pairs (f^{l_{n}}(x), f^{l_{n}−m_{n}}(c_{n})) arise from the pull-backs of U_{i(n)} hitting a recurrent critical point c_{n} for the first time. Unlike before, in this case we rely upon topological and dynamical properties of dendrites established in Section 4. Observe first that by the choice of ε the point c_{n} must belong to D. Also, by the choice of U_{i(n)} we know that it has diameter less than ε and contains some points of limit sets of recurrent critical points of f. Therefore Lemma 4.3 applies to U_{i(n)} and all pull-backs of U_{i(n)} will have connected intersections with D. In particular, if W is the (n − m_{n})-pull-back of U_{i(n)} corresponding to c_{n}, then W ∩ D is connected. Therefore there exists a unique arc I_{n} connecting f^{m_{n}}(x) and c_{n} inside W ∩ D. Observe that I_{n} has the endpoints c_{n} and f^{m_{n}}(x) which both belong to J(f). Observe also that f^{m_{n}}(I_{n}) ⊂ U_{i(n)} which implies that f^{m_{n}}(I_{n}) contains no attracting or parabolic periodic points (otherwise the choice of points in the Basic Setup for f^{n}(x) would have been different by the definition). Hence Lemma 4.6 applies to the just constructed sequence of arcs I_{n} (recall that f has no Cremer points). It implies that ω(x′) = ω(c′) as desired.

Observe that along the way we establish one extra property of our construction. Namely, if c is a non-recurrent critical point which appears in the Basic Setup infinitely many times, then it follows from Theorem 3.5 that d(f^{m_{n}}(x), c) → 0 for m_{n}’s corresponding to c_{n} = c in Basic Setup. Therefore L_{c} = {c} for any such critical point c.

Let us use this to complete the proof. Indeed, as we have just shown the point x is followed by points of A. Hence by Theorem 2.1 there exists a point c ∈ A such that ω(c) = ω(x) with all the properties listed in Theorem 2.1. We assume that x is not preparabolic. Then c cannot be a parabolic periodic point because it is known that ω(x) can be a parabolic cycle if and only if x is preparabolic. Hence c is either recurrent or non-recurrent critical point. Suppose that c is a non-recurrent critical point. Since by Theorem 2.1 c appears in Basic Setup infinitely many times, we see by the preceding paragraph that L_{c} = {c}. If the bounded case takes place, then for some m we have f^{m}(x) = c, which is impossible since x is not precritical. Hence the unbounded case takes place. By Theorem 2.1 c is then recurrent, because it belongs to L_{c}, a contradiction with the assumption that c is non-recurrent. Thus c is recurrent and so ω(x) = ω(c) for some recurrent critical point c.

Together with Lemma 5.1 this theorem immediately implies our main Theorem 5.4.

Theorem 5.4. For a dendrite-critical polynomial f and a conformal measure μ one of the following holds.

(1) For μ-almost every x ∈ J(f), ω(x) = J(f).

(2) For μ-almost every x ∈ J(f), ω(x) = ω(c(x)) for some critical point c(x) depending on x, and either x is an eventual preimage of c(x), or x is preparabolic, or c(x) can be chosen to be recurrent.

An immediate corollary is the following.

Corollary 5.5. If f is a dendrite-critical polynomial and μ is a non-atomic conformal measure, then for μ-a.e. point x the set ω(x) contains a critical point.

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Department of Mathematics, University of Alabama in Birmingham, University Station, Birmingham, AL 35294-2060

   E-mail address: ablokh@math.uab.edu

Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, IN 46202-3216

   E-mail address: mmisiure@math.iupui.edu