ONE-DIMENSIONAL FIBONACCI TILINGS AND DERIVATIVES OF TWO-COLOUR ROTATIONS OF A CIRCLE

VLADIMIR ZHURAVLEV

Abstract. This is a study of the two-colour (composite) rotations S_{ε}(a, b), a, b ∈ ℤ, of the unit circle C that has a coloration which depends on the continuous parameter ε ∈ C. We prove that these rotations S_{ε}(a, b) are connected with one-dimensional Fibonacci tilings T il(ε_{m}) of the unit interval I ≃ ℝ/ℤ and S_{ε}(a, b) have interesting global and local structures.

Global structure. The space of the two-colour rotations S_{ε}(a, b) is closed under the operation of boundary differentiations or the restrictions of rotations on intervals in the Fibonacci tilings T il(ε_{m}). By this invariance, self-similarity properties of the two-colour rotations follow for quadratic irrationals. It means that the derivative rotation is equal to a renormalization-rotation.

Local structure. Among the results here are following. The attractors Att(S_{ε}(a, b)) ⊂ C preserve their measure for the parameter ε in open arcs. It implies that the function ν_{ε}(θ) of the frequency distribution of points in the S_{ε}(a, b)-orbits has a plateau. The circle C can be split into open arcs C′ ⊂ C so that the C′-restriction of the rotation S_{ε}(a, b) is isomorphic to an integral transformation of a simple rotation or a two-colour rotation of C.

Dedicated to Anatoli Nikolaevich Andrianov

Contents

1. Introduction

2. One-dimensional Fibonacci tilings 2.1. Branching process 2.2. The branching process and tilings 2.3. Global coordinates

3. Fibonacci numbers and the golden section 3.1. Periodicity of Fibonacci numbers moda 3.2. Fibonacci numbers and the Euler function 3.3. Distribution of two points on a circle

4. Boundary derivatives and self-similarity 4.1. Derivatives of the simple shift 4.2. Upper derivatives of two-colour shifts 4.3. Self-similarity of the composite shift 4.4. Lower derivatives of the composite shift

5. Internal derivatives and plateaus 5.1. Lacunas 5.2. Spirals 5.3. Frequency distribution and plateaus

6. Spectral decompositions and singularities 6.1. The composite shift S_{ε}(3, 1)

1Date: 15 June, 2004

6.2. Spectrum 6.3. Singularities References

1. Introduction

1. Composite shifts and one-dimensional Fibonacci tilings. Let’s identify the interval I = [0, 1) with the unit circle C ≃ ℝ/ℤ. Consider a simple shift S : x ↦ x + τ mod 1, where τ = (1 + \sqrt{5})/2 = 1.618 … is the golden section. We define derivatives (induced transformations, the first return maps)

S^{(m)} = S |I^{m}

as restrictions of the shift S on some intervals I^{m} = [ε_{m}, 1) ⊂ I. Assume that the derivative transformations

S^{(m)} ∼ S^{±1}  (1.1)

are similar to the shift S, or the inverse shift S^{−1} : x ↦ x−τ mod 1. Then we arrive at a concept of the branching B-process (see Section 2). This process generates the eigenvalues

ε_{m} = 1 − τ^{−m}  (1.2)

with numbers m = 0, 1, 2, … The property (1.1) exists whenever the parameter ε_{m} in I^{m} = [ε_{m}, 1) has form (1.2). The power ±1 in S^{±1} depends on the fact whether the number m is even or odd. Each differentiation interval

I^{m} = [ε_{m}, 1)

is split

I^{m} = G_{0}^{m} ⊕ E_{0}^{m}

into intervals G_{0}^{m}, E_{0}^{m} so that they touch each other. Their lengths g_{m} = |G_{0}^{m}|, e_{m} = |E_{0}^{m}| are equal to

g_{m} = 1 − ε_{m+2} = τ^{−(m+2)}, e_{m} = 1 − ε_{m+1} = τ^{−(m+1)}

for even m, and

e_{m} = 1 − ε_{m+2} = τ^{−(m+2)}, g_{m} = 1 − ε_{m+1} = τ^{−(m+1)}

for odd m. In the case of even m, suppose S shifts the intervals G_{0}^{m}, E_{0}^{m} F_{m+1}−1 and F_{m+2} − 1 times, respectively, where F_{m} are the Fibonacci numbers:

F_{m+2} = F_{m+1} + F_{m}

and F_{1} = F_{2} = 1 Then we get the one-dimensional Fibonacci tiling T il(ε_{m}) of the unit interval I of the level m. For odd m, the number of shifts is equal to F_{m+2} − 1 and F_{m+1} − 1, respectively.

The following invariancy property

S(T il(ε_{m})) = T il^{+}(ε_{m})  (1.3)

of the Fibonacci tilings is basic. The dual tiling T il^{+}(ε_{m}) of the unit interval I is received from T il(ε_{m}) by the rearrangement I^{m} = E_{0}^{m} ⊕G_{0}^{m} of its intervals G_{0}^{m} and E_{0}^{m}. From invariancy (1.3) we get self-similarity property (1.1) for the derivatives S^{(m)}.

We can construct the one-dimensional Fibonacci tilings T il(ε_{m}) using the following methods [10]:

1) The branching B-process.

2) Substitutions or the division of long intervals in ratio τ (inflation rules).

3) The recurrence formula (see Subsection 5.3)

T il^{+}(ε_{m+2}) = τ^{−1} · \overline{T il^{+}(ε_{m+1})} ⊕ τ^{−2} · T il^{+}(ε_{m}),  (1.4)

where the line over \overline{T il^{+}(ε_{m+1})} indicates the inversion of designations for long and short intervals. We denote by ⊕ the non-commutative operation that puts the first tiling to the second tiling compressed by τ^{−1} and τ^{−2} times, respectively.

4) The quasi-local coordinates (see Subsection 5.1).

5) The global coordinates (see Subsection 2.3).

6) The differential approach (in this article).

A two-colour shift or a composite shift S_{ε}(a, b) : I ⟼ I with integer parameters a, b ∈ ℤ depends on the unit interval coloring

I = I_{ε}^{−} ⊕ I_{ε}^{+},  (1.5)

where I_{ε}^{−} = [0, ε), I_{ε}^{+} = [ε, 1). Then the composite shift S_{ε}(a, b) is determined by the two-colour vector field (cf. [1])

x ⟼ x + aτ mod 1, if x ∈ I_{ε}^{−},

x ⟼ x + bτ mod 1, if x ∈ I_{ε}^{+}. (1.6)

Our goal in this paper is to research geometrical and dynamic properties of the composite shifts S_{ε}(a, b) and to describe their structure in terms of the onedimensional Fibonacci tilings T il(ε_{m}).

2. Derivatives of composite shifts. Here we use a method of differentiation which means a restriction of the composite shifts S_{ε}(a, b) on special intervals I^{m}, I_{m}, and J^{m} connected with the Fibonacci tilings T il(ε_{m}). We consider two kinds of derivatives :

1) boundary upper derivatives and boundary lower derivatives,

2) internal derivatives.

The upper derivatives of the order m = 0, 1, 2, …

S_{ε}(a, b)^{(m)} = S_{ε}(a, b) |I^{m} × I^{m}  (1.7)

are restrictions of the composite shift S_{ε}(a, b) for x and ε both on the intervals I^{m} = [ε_{m}, 1), where ε_{m} are eigenvalues (1.2) of the B- process.

Note that the parameters a and b in definition (1.6) of the composite shift S_{ε}(a, b) are symmetric. Therefore, we can compare the upper derivatives S_{ε}(a, b)^{(m)} (1.7) with their dual lower derivatives

S_{ε}(a, b)_{(m)} = S_{ε}(a, b) |I_{m} × I_{m},

where the differentiation intervals I_{m} = inv(I^{m}) = (0, 1−ε_{m}] are symmetric to the intervals I^{m} under the involution inv : x ↦ 1 − x of the unit interval I = [0, 1).

For the composite shift S_{ε}(a, b), we define internal derivatives by the following local and global ways:

S_{ε}(a, b)|I^{m}(δ) × J^{m},  S_{ε}(a, b)|Orb(I^{m}(δ)) × J^{m}.  (1.8)

Here I^{m}(δ) = I^{m} + δ mod 1,

δ = |E_{0}^{m}|, J^{m} = [1 − |G_{0}^{m}| − |L^{m}|, 1 − |G_{0}^{m}|),

if the level m is even, and

δ = ε − (1 − |E_{0}^{m}|), J^{m} = [1 − |E_{0}^{m}| − |L^{m}|, 1 − |E_{0}^{m}|)

in the case of odd m . We denote by Orb(I^{m}(δ)) the orbit of the interval I^{m}(δ) under the shift S_{ε}(a, b) and by L^{m} a lacuna of the orbit Orb(I^{m}(δ)) (see Subsection 5.1).

The space S of two-colour shifts (1.6) is closed under the operation of boundary differentiations (Theorem 1 and Theorem 2). Assume that the composite shift S_{ε}(a, b) (1.6) has the integer parameters a, b with the same sign and the Fibonacci numbers F_{m+1}, F_{m+2} satisfy the congruences

F_{m+1} ≡ F_{m+2} ≡ F mod a,

where 1 ≤ F ≤ |a|. Suppose |b| ≤ s((|b|F^{−1})_{a}, m) and let s(k, m) be a function from Subsection 4.2. Then the upper derivative of the order m

S_{ε}(a, b)^{(m)} ∼ S_{ε′} ((−1)^{m}a, (−1)^{m}sign(b) · b′)  (1.9)

is a composite shift once again. The derivative shift S_{ε}(a, b)^{(m)} is similar to the shift on the right-hand side in formula (1.9), where b′ = (|b|F^{−1})_{a} is equal to the least positive solution of the congruence xF ≡ |b| mod a and ε′ = (ε − ε_{m})/(1 − ε_{m}) is a renormalisation of the parameter ε. The similarity ∼ of the composite shifts in (1.9) means the conjugacy

S_{ε}(a, b)^{(m)} = h_{m} ∘ S_{ε′}((−1)^{m}a, (−1)^{m}sign(b) · b′) ∘ h_{m}^{−1}

under the affine isomorphism h_{m}(x) = τ^{−m}(x − 1) + 1 of the unit interval I and the differentiation interval I^{m}.

For the lower derivative S_{ε}(a, b)_{(m)} we have the symmetric similarity (cf. (1.9))

S_{ε}(a, b)_{(m)} ∼ S_{ε′}((−1)^{m}sign(a) · a′, (−1)^{m}b)).

The internal derivatives (1.8) of two-colour shifts have a local non-invariant characteristic. Let ν_{ε} be a function of frequency distribution of S_{ε}(a, b)-orbits. Here we consider an elementary scenario that explains how plateaus of the function ν_{ε} can arise.

Let S_{ε}(a, b) be composite shift (1.6) with integer coprime parameters 1 ≤ b < a and let a, b satisfy the congruences

F_{m+1} ≡ b mod a, F_{m+2} ≡ 0 mod a.

Then we have the following isomorphisms

\begin{matrix}S_{ε}(a, b)| I^{m}(δ) × J^{m} ≃ S^{±1}, \\ S_{ε}(a, b)|Orb(I^{m}(δ)) × J^{m} ≃ ∫ S^{±1} d_{ε}, \end{matrix}  (1.10)

with S^{+1} or S^{−1} for even or odd m, respectively. In (1.10), ∫ S^{±1} d_{ε} is an integral transformation of the simple shift S^{±1} : x ⟼ x ± τ mod 1 with respect to a two-colour distribution d_{ε} on the unit interval I.

3. Self-similarity of composite shifts. There is a self-semilarity phenomenon for composite shifts. It follows from two facts:

1) the space S of composite shifts S_{ε}(a, b) (1.6) is closed under the boundary derivatives;

2) the golden section τ is a quadratic irrational number. Since the upper derivatives S_{ε}(a, b)^{(m)} and the lower derivatives S_{ε}(a, b)_{(m)} are dual to each other, we formulate some consequences from invariancy (1.9) only for the upper derivatives.

Suppose the integer parameters a, b have the same sign and satisfy the condition 1 ≤ |b| ≤ |a|. Then the composite shift S_{ε}(a, b) has the self-similarity property (see Corollary 1)

S_{ε}(a, b)^{(m)} ∼ S_{ε′}(a, b).  (1.11)

The order m in (1.11) is equal to T or 2T, if the period T = T (a) of the sequence of the Fibonacci numbers F_{1}, F_{2}, …, F_{n}, … mod a is even or odd, respectively.

Let the parameter a be a prime number p satisfying the condition 1 ≤ b ≤ p. Then self-similarity property (1.11) becomes as follows (see Corollary 2):

S_{ε}(p, b)^{(p−1)} ∼ S_{ε′}(p, b)  (1.12)

for prime numbers p ≡ 1 or 4 mod 5,

S_{ε}(p, b)^{(2(p+1))} ∼ S_{ε′}(p, b)

for prime numbers p ≡ 2 or 3 mod 5, and

S_{ε}(p, b)^{(20)} ∼ S_{ε′}(p, b)

for p = 5.

Consider the case when a is equal to the Fibonacci number F_{m} (see Corollary 3). In this case we have

S_{ε}(F_{m}, b)^{(2m)} ∼ S_{ε′}(F_{m}, b)  (1.13)

for even m, and

S_{ε}(F_{m}, b)^{(4m)} ∼ S_{ε′}(F_{m}, b)

for odd m. For m = 2, we have a commutative diagram

\begin{aligned}S_{ε}(1, 1)^{(2)} & ∼ & S_{ε′}(1, 1) \\ ↓≃ & & ↓≃ \\ S^{(2)} & ∼ & S \end{aligned}  (1.14)

In particular, from (1.14) it follows the derivative formula S^{(2)} ∼ S (1.1) for the simple shift S.

For m = 3, we obtain the first non-trivial case of the self-similarity property for composite shifts (see Fig. 1)

S_{ε}(2, 1)^{(6)} ∼ S_{ε′}(2, 1).  (1.15)

Fig. 1. The attractor development Att(2, 1).

In more general case, the composite shift S_{ε}(a, b) is a contracting mapping

I ⊃ S_{ε}(a, b)(I) ⊃ ⋯ ⊃ S_{ε}^{k}(a, b)(I) ⊃ ….

For any integer parameters a, b , ab > 0, the above chain contains only a finite number of strict inclusions. The attractor Att(S_{ε}(a, b)) of the composite shift S_{ε}(a, b) is defined as the intersection

Att(S_{ε}(a, b)) = ⋂_{k=0}^{∞} S_{ε}^{k}(a, b)(I),

where we denote by S_{ε}^{k}(a, b) the k-multiple composition of the shift S_{ε}(a, b). The unit interval I splits into

I = Att(S_{ε}(a, b)) + Spir_{ε}  (1.16)

the attractor Att(S_{ε}(a, b)) and a spiral Spir_{ε}. The attractor Att(S_{ε}(a, b)) is the minimal set of the shift S_{ε}(a, b) and the spiral Spir_{ε} is an area any point of which gets in the attractor Att(S_{ε}(a, b)) through a finite number of iterations. We introduce the dynamic metric dis(ε) as the number of iterations. Then we can say that dis(ε) < ∞, i.e., the maximal distance between the spiral Spir_{ε} and the attractor Att(S_{ε}(a, b)) is finite.

For the composite shift S_{ε}(2, 1) the following estimations are known:

\frac{c_{1}}{1 − ε} − 2 < dis(ε) < \frac{c_{2}}{1 − ε} + 1

with the constants c_{1} = \frac{τ^{2}}{2\sqrt{5}} ≈ 0.58... and c_{2} = τc_{1} ≈ 0.94... In particular, from

these estimations it follows that

dis(ε) ⟶ ∞ as ε ↑ 1.

For the shift S_{ε}(2, 1), Fig. 1 represents the development of decomposition (1.16) through the parameter ε ∈ I

I × I = Att(2, 1) + Spir.  (1.17)

Accoding to the unit interval coloration I = I_{ε}^{−} ⊕ I_{ε}^{+} defined in (1.5), the development Att(2, 1) of the attractor Att(S_{ε}(2, 1)) consists of the grey Att(2, 1)^{−} areas and the dark Att(2, 1)^{+} areas, and the development Spir of the spiral Spir_{ε} corresponds to the light area.

Self-similarity (1.15) of the composite shift S_{ε}(2, 1) (see Fig. 1) means that the restriction of development (1.17) on the upper right square I^{6} × I^{6} is similar to complete development (1.17). So the attractor development Att(2, 1) is a fractal.

4. Internal derivatives and plateaus. The following equality

Orb(I^{m}(δ)) = Att(S_{ε}(a, b))

from Subsection 5.3 is true for all parameters ε ∈ J^{m}. It clears up a sense of the second formula in (1.10). Using the above equality, we obtain an isomorphism

S_{ε}(a, b) |Att(S_{ε}(a, b)) × J^{m} ≃ ∫ S^{±1}d_{ε}  (1.18)

from which it follows the distribution properties of points in the S_{ε}(a, b) - orbits (see Theorem 3).

Suppose the composite shift S_{ε}(a, b) satisfies condition (1.10) and its parameter ε belongs to the interval J^{m}. Then for any x, θ in I = [0, 1) there exists the limit

ν_{ε}(θ) = lim_{n→∞} \frac{1}{n} ♯\{k : S_{ε}^{k}(a, b)(x) ≤ θ, k = 0, 1, …, n − 1\}

which is independent of the choice of the initial point x. The function ν_{ε}(θ) is equal to the frequency when points in the S_{ε}(a, b)-orbit of x visit the interval [0, θ]. For ν_{ε}(θ) we prove the formula (see Theorem 3)

ν_{ε}(θ) = \frac{|Att(S_{ε}(a, b)) ∩ [0, θ]|}{|Att(S_{ε}(a, b))|},  (1.19)

where we denote by | · | a Haar measure on the torus C ≃ ℝ/ℤ.

A diagonal frequency ν_{ε} is defined as the specialization ν_{ε} = ν_{ε}(θ) with θ = ε. For the diagonal frequency ν_{ε} formula (1.19) has the form

ν_{ε} = \frac{|Att(S_{ε}(a, b))^{−}|}{|Att(S_{ε}(a, b))|},  (1.20)

where Att(S_{ε}(a, b))^{−} is the intersection of the attractor Att(S_{ε}(a, b)) and the interval I_{ε}^{−} = [ 0, ε).

The grey area Att(2, 1)^{−} in Fig. 1 represents the development of the set Att(S_{ε}(2, 1))^{−} through the parameter ε ∈ I. The existence of plateaus for the diagonal frequency ν_{ε}(see Fig. 2) follows from formula (1.20). It means that the frequency ν_{ε} is independent of the parameter ε, if ε ∈ J^{m} ( see Proposition 5).

Let S_{ε}(F_{n}, F_{n−1}) be composite shift (1.6) with the parameter ε ∈ J^{m} , where m ≡ n − 1 mod T (F_{n}), n ≥ 3, and T (F_{n}) is the period of the sequence of the Fibonacci numbers F_{1}, F_{2}, …, F_{m}, … mod F_{n}. Then for any initial point x in I = [0, 1) the frequency ν_{ε}, when points in the S_{ε}(F_{n}, F_{n−1})-orbit of x visit the interval I_{ε}^{−} = [0, ε), is equal to (see Proposition 5)

ν_{ε} = \frac{1 − F_{n−1}τ^{−(m+2)}}{1 + F_{n−2}τ^{−(m+2)}}  (1.21)

or

ν_{ε} = 1 − \frac{τ^{−(m+2)}}{|Att(S_{ε}(F_{n}, F_{n−1}))|}.  (1.22)

Formula (1.22) connects the frequency ν_{ε} with the measure |Att(S_{ε}(F_{n}, F_{n−1}))| of the attractor Att(S_{ε}(F_{n}, F_{n−1})). Thus, if it is known the frequency ν_{ε}, we can find the measure |Att(S_{ε}(F_{n}, F_{n−1}))| using formula (1.22).

Some corollaries follow from formulas (1.21) and (1.22).

1. Under the above conditions the attractor measure |Att(S_{ε}(F_{n}, F_{n−1}))| is equal to

|Att(S_{ε}(F_{n}, F_{n−1}))| = \frac{τ^{−(m+2)}}{1 − ν_{ε}} .

So

|Att(S_{ε}(F_{n}, F_{n−1}))| → \frac{1}{F_{n}}

as ε → 1 and ε lies in the union of the intervals J^{m} (see (1.8)).

2. Suppose the parameter ε of the composite shift S_{ε}(F_{n}, F_{n−1}), n ≥ 3, belongs to the interval J^{n−2} = [τ^{−(n−1)}, 1 − τ^{−n}). Then the frequency ν_{ε} is independent of n and is equal to

ν_{ε} = τ^{−1} = 0.618 …

3. Suppose the parameter ε of the composite shift S_{ε}(2, 1) belongs to the interval J^{m} = [1 − τ^{−m}, 1 − τ^{(m+2)}) of the level m ≡ 1 mod 3, m ≥ 1. Then the frequency ν_{ε} is equal to (see Fig.2)

ν_{ε} = th_{τ} (\frac{m}{2} + 1).  (1.23)

Here the function th_{τ} (x) = (τ^{x} − τ^{−x})/(τ^{x} + τ^{−x}) is a hyperbolic tangent with the basis τ = (1 + \sqrt{5})/2.

Fig. 2. The diagonal frequency ν_{ε} for the two-color shift S_{ε}(2, 1)

5. Spectral decompositions and singularities. Let μ(ν) be a measure of ε ∈ I so that the frequency ν_{ε} is equal to ν, i.e.,

μ(ν) = |{ε ∈ [0, 1) : ν_{ε} = ν}|.

The spectrum sp(a, b) of the composite shift S_{ε}(a, b) is the set of all ν ∈ I satisfying the inequality μ(ν) > 0, where μ(ν) is the weight of ν.

For the shift S_{ε}(2, 1), spectrum sp(2, 1) consists of the points (see Fig. 2)

ν_{ε} = th_{τ} (\frac{m}{2} + 1)  (1.24)

with the weights μ^{m} = τ^{−(m+1)} and the numbers m ≡ 1 mod 3, m ≥ 1. A plateau of the frequency function ν_{ε} corresponds to each spectral point ν^{m} ∈ sp(2, 1). The weight μ^{m} = μ(ν^{m}) is equal to the length of an interval, where the function ν_{ε} is constant. Since

ν^{m} ∼ 1 − τ^{−(m+2)}

as m → ∞, the graph of the function μ(ν) is asymptotically self-similar with center at the point (ν, μ) = (1, 0) and ratio τ^{−3}.

We write the spectrum sp(2, 1) of the composite shift S_{ε}(2, 1) as

sp(2, 1) = ( |).  (1.25)

0  ν^{1}  ν^{4} ν^{7} 1 μ^{7} μ^{4} μ^{1}^{0.4}

Fig. 3. The spectrum S_{ε}(2, 1)

In (1.25), the symbol ( |) means that the shift S_{ε}(2, 1) has an elementary spectrum (see Fig. 3) generated by a unique point ν^{1} = τ^{−1} = 0.618.... Moreover, the symbol ( |) means that the spectrum sp(2, 1) is asymptoticaly self-similar and the line | is its fundamental domain.

Fig. 5 represents the spectrum of the composite shift S_{ε}(3, 1). The development Att(3, 1) of the attractor Att(S_{ε}(3, 1) is shown in Fig. 4.

Fig. 4. The attractor development Att(3, 1)

0  1 ν μ 0.2 0.1

Fig. 5. The spectrum S_{ε}(3, 1)

In the same way as for the spectrum sp(2, 1) (1.25),we introduce the following notation for the spectrum sp(3, 1):

sp(3, 1) = ((−2, −1)|(−1, −2) |),  (1.26)

where

sp(−2, −1) = (| (2, 1)).  (1.27)

The involution inv : ν → 1 − ν maps the spectrum sp(2, 1) to the spectrum sp(−1, −2). Thus, a 180^{∘} rotation maps the development Att(2, 1) to the development Att(−1, −2) (cf. Fig. 1 and Fig. 4). In spectral decomposition (1.26), the first vertical line | corresponds to a limiting point in the spectrum sp(3, 1). The second vertical line ⌣ | (with the space on the left) corresponds to its discrete point.

The composite shifts S_{ε}(a, b) with the integer coprime parameters a, b satisfying the condition 1 ≤ b < a ≤ 4 have the following spectral decompositions:

\begin{matrix}sp(3, 1) = (| (2, 1)|(−1, −2) |), \\ sp(3, 2) = (−1, −2) | (3, 1), \\ sp(3, 2) = ((−1, −2) | (2, 1)|), \\ sp(4, 1) = ((2, 1)|(−1, −2) |), \\ sp(4, 3) = ((2, 3)|(−1, −2) | (2, 1)|(3, 2)|). \end{matrix}  (1.28)

Fig. 6. The attractor developments for two-color shifts (a) Att(−2, −1), (b) Att(3, 2), (c) Att(4, 1), (d) Att(4, 3)

Spectral decompositions (1.28) show that the composite shifts S_{ε}(a, b) are decomposed into the sum of elementary transformations. Each transformation is isomorphic to

1) an integral of the simple shift S^{±1} : x ↦ x ± τ mod 1 or

2) an integral of the two-colour shift S_{ε}(2, 1) or its symmetric shift S_{ε}(−1, −2). So there exist only two basic shifts S and S_{ε}(2, 1).

The singularities are the limiting points s of the spectrum sp(a, b) in torus topology I ∼ ℝ/ℤ. For every such point s we define the degree deg(s). In the spectrum sp(2, 1) (see Fig. 3) the singularity s ≡ 0 mod 1 has degree 1, and in the spectrum sp(3, 1) (see Fig. 5) the singularity s = 0.809 … has degree 1 and s ≡ 0 mod 1 has degree 2. By definition, discrete points of a spectrum have zero degree. The degree of the whole spectrum sp(a, b) is equal the maximal degree of its singularities.

The composite shifts S_{ε}(a, b) from list (1.28) are distributed according to their degrees deg(a, b) as follows:

deg(2, 1) = 1, deg(−2, −1) = 1,
deg(3, 1) = 2, deg(3, 2) = 2, deg(4, 1) = 2,
deg(4, 3) = 3.

Thus, the degree deg(a, b) of the spectrum sp(a, b) specifies a complexity level of the two-colour shift S_{ε}(a, b).

In accordance with Proposition 6, for any n = 1, 2, 3, … there exists a composite shift S_{ε}(a, b) with the integer coprime parameters 1 ≤ b < a ≤ F_{n+2} so that the spectral degree deg(a, b) satisfies the inequality

deg(a, b) ≥ n.

6. Around the Fibonacci tilings. There are several methods for building of the one-dimensional Fibonacci tilings T il(ε_{m}) and their limiting tiling

T il(ε_{∞}) = lim_{m→∞} τ^{m}T il(ε_{m})

(see, for example, [2], [6], [10]). In [10] one can find the following methods: the B-process (the branching process), the substitution method (inflation rules), the recurrence Fibonacci method (see (1.4)), the quasi-local method, and the global method.

This article is based on the differential approach (see (1.1), (1.3) ). We build the tilings T il(ε_{m}) by using the derivatives S^{(m)} = S |I^{m} of the simple shift S : x ↦ x + τ mod 1 of the unit interval I = [0, 1). The tilings T il(ε_{m}) appear if we consider the following problem:

Let the derivative transformations S^{(m)} = S |I^{m} be similar S^{(m)} ∼ S^{±1} to the shift S or its inverse shift S^{−1} (in more general case, we can assume that S^{(m)} is similar to some shift of the unit interval I). What is a differential interval I^{m} ⊆ I?

In the context of number theory derivatives appeared implicitly in the study of regularities of distribution for fractional parts ⟨nα⟩, where n = 1, 2, 3, … and α is irrational. An interval J ⊆ I is called a bounded remainder set for the sequence ⟨nα⟩, if the sequence of remainders

r_{α,J}(N) = ♯{n < N : ⟨nα⟩ ∈ J} − N | J |

is bounded as N → ∞. It is known [3]-[5] that the interval J is a bounded remainder set for ⟨nα⟩ if and only if the length | J |> 0 lies in αℤ + ℤ. Since, by (1.2), the length | I^{m} | is equal to τ^{−m} = (−1)^{m}(F_{m+1} − F_{m}τ) that lies in the set τℤ + ℤ then the interval I^{m} is a bounded remainder set for the sequence ⟨nτ⟩. Moreover, any bounded remainder interval J is split into the differentiation intervals I^{m}.

In this article we discuss a connection between the Fibonacci tilings T il(ε_{m}) and the two-colour (composite) shifts S_{ε}(a, b) of the interval I = [0, 1) (see (1.6)). The tilings T il(ε_{m}) are used to research the derivative shifts S_{ε}(a, b)|∗ (see (1.7), (1.8)).

The composite shifts S_{ε}(a, b) by themselves have arisen from the two-dimensional version [8], [9] of the border layer of quasicrystal growth.

Acknowledgment. First of all, I wish to thank Prof. Yu.I. Manin of Max-Planck-Institut fu^{..}r Mathematik (Bonn, Germany), where big parts of the research were carried out and reported, for his kind invitation.

Especially I should like to thank Prof. A.N. Andrianov and Prof. Siegfried Boecherer for support of my scientific work.

I am grateful to N. Manuilov, A. Shutov and V. Savina who helped me to prepare this paper.

The research was partly supported by Russian Fund of Fundamental Researches (Grant 03-01-00368).

2. One-dimensional Fibonacci tilings

2.1. Branching process. We consider a unit circle C and identify it with the interval I = [0, 1), i.e. I ≃ ℝ/ℤ. Now we introduce the concept of branching Bprocess on the interval I. It is defined in [10] by the following conditions:

1) Let P = {ε_{0}, ε_{1}} be a set of the marked points ε_{0} = 0, ε_{1} = 2 − τ.

2) A point ε runs through the interval I with a positive constant speed. Its movement begins from the point ε_{0}.

3) The images ε^{α} of each marked point ε_{α} begin movement together with the point ε with the same speed.

4) As soon as one of these points ε^{α} reaches one of the marked points ε_{α}, we get a new set P of marked points. The set P contains all the points ε^{α} and old marked points.

5) The point ε continues its movement after each branching.

From the definition of the B-process it follows that the set of the marked points P = P (ε) depends on ε and the set P (ε) grows for a discrete set of the parameter’s values ε ∈ I :

ε_{0} < ε_{1} < ⋯ < ε_{m} < …,  (2.1)

where ε_{0} = 0. The set E = {ε_{m}} is a spectrum of the B-process and ε_{m} are the eigenvalues of the B-process.

ε_{0}  ε_{1}  ε_{2}  ε_{3} ε_{4} ε_{5} Fig. 2.1. The spectrum of B-process

The spectrum E = {ε_{m}} (2.1) is infinite. The eigenvalues ε_{m} satisfy the following equalities and recurrence formulaes:

\begin{matrix}ε_{m} = ⟨(−1)^{m}F_{m} · τ⟩ = 1 − (τ − 1)^{m} = 1 − τ^{−m}, \\ ε_{m+2} = ε_{m} − ε_{m+1} + 1 \end{matrix}  (2.2)

for m = 0, 1, 2, …

2.2. The branching process and tilings. The set of the marked points P (ε) tiles the unit interval into small intervals . Let T il(ε) be an appropriate tiling of I on left-closed, right-open intervals [∗, ∗) ⊂ I. For ε = ε_{m} the tiling

T il(ε_{m}) = ∑_{j} G_{j}^{m} + ∑_{i} E_{i}^{m}  (2.3)

is the eigen Fibonacci tiling. It consists of intervals of two kinds G_{j}^{m} and E_{i}^{m} which have lengths g_{m} and e_{m} , respectively.. The quotient of these lengths is equal to

\begin{matrix}\frac{e_{m}}{g_{m}} = τ, m = 0, 2, 4, …, \\ \frac{g_{m}}{e_{m}} = τ, m = 1, 3, 5, … \end{matrix}  (2.4)

We see that the quotient of lengths depends on the fact whether the level m of the tiling T il(ε_{m}) is even or odd. The spectrum E = {ε_{m}} and the eigen tilings T il(ε_{m}) are connected by the formula

ε_{m+1} = ε_{m} + min{g^{m}, e^{m}}.  (2.5)

Here m ≥ 1 and ε_{1} = 2 − τ. Moreover, we have the convenient formulas for the lengths g_{m}, e_{m} of the intervals G_{j}^{m}, E_{i}^{m}:

g_{m} = 1 − ε_{m+2} = τ^{−(m+2)}, e_{m} = 1 − ε_{m+1} = τ^{−(m+1)}  (2.6)

for even levels m, and

g_{m} = 1 − ε_{m+1} = τ^{−(m+1)}, e_{m} = 1 − ε_{m+2} = τ^{−(m+2)}  (2.7)

for odd m.

If ε is not an eigenvalue, the tiling T il(ε) is not eigen. If m is even and ε_{m} ≤ ε ≤ ε_{m+1}, we can construct the tiling

T il(ε) = ∑_{j} G_{j}^{m} + ∑_{i} E_{i}^{m}  (2.8)

using the eigen tiling T il(ε_{m}) (2.3) by the help of the substitutions

G_{j}^{m} = GG_{j}^{m}(ε) ⊕ G_{j}^{m}(ε), E_{i}^{m} = GE_{i}^{m}(ε) ⊕ E_{i}^{m}(ε).

Here the left intervals have length |GG_{j}^{m}(ε)| = |GE_{i}^{m}(ε)| = ε − ε_{m}. Hence, for ε = ε_{m} the tiling T il(ε) (2.8) coincides with the eigen tiling T il(ε_{m}) (2.3). At ε = ε_{m+1} the tiling T il(ε) passes to the next eigen tiling

T il(ε_{m+1}) = ∑_{j} G_{j}^{m+1} + ∑_{i} E_{i}^{m+1},  (2.9)

where

\begin{matrix}∑_{j} G_{j}^{m+1} = ∑_{j} G_{j}^{m}(ε_{m+1}) + ∑_{i} GE_{i}^{m}(ε_{m+1}), \\ ∑_{i} E_{i}^{m+1} = ∑_{i} E_{i}^{m}(ε_{m+1}). \end{matrix}

If the level m is odd, we can obtain tiling T il(ε) (2.8) by the substitutions

G_{j}^{m} = EG_{j}^{m}(ε) ⊕ G_{j}^{m}(ε), E_{i}^{m} = EE_{i}^{m}(ε) ⊕ E_{i}^{m}(ε)

Here |EG_{j}^{m}(ε)| = |EE_{i}^{m}(ε)| = ε − ε_{m}. At ε = ε_{m+1} eigen tiling T il(ε_{m+1}) (2.9) consists of intervals

\begin{matrix}∑_{j} G_{j}^{m+1} = ∑_{j} G_{j}^{m}(ε_{m+1}), \\ ∑_{i} E_{i}^{m+1} = ∑_{j} EG_{j}^{m}(ε_{m+1}) + ∑_{i} EE_{i}^{m}(ε_{m+1}). \end{matrix}

The above construction describes continuous reorganization of the eigen tilings T il(ε_{m}) to the next level tilings T il(ε_{m+1}). F ig.2.2 presents the reorganisation process for the eigen Fibonacci tilings T il(ε_{m}), m = 0, …, 4.

[figure: ε_{0}; G_{0}^{0}; E_{0}^{0}; T il(ε_{0}); ε_{1}; G_{1}^{1}; G_{0}^{1}; E_{0}^{1}; T il(ε_{1}); ε_{2}; E_{2}^{2}; G_{1}^{2}; E_{1}^{2}; G_{0}^{2}; E_{0}^{2}; T il(ε_{2}); ε_{3}; G_{2}^{3}; E_{2}^{3}; G_{4}^{3}; G_{1}^{3}; E_{1}^{3}; G_{3}^{3}; G_{0}^{3}; E_{0}^{3}; T il(ε_{3}); ε_{4}; E_{5}^{4}; G_{2}^{4}; E_{2}^{4}; E_{7}^{4}; G_{4}^{4}; E_{4}^{4}; G_{1}^{4}; E_{1}^{4}; E_{6}^{4}; G_{3}^{4}; E_{3}^{4}; G_{0}^{4}; E_{0}^{4}; T il(ε_{4}); Fig. 2.2. Fibonacci tilings T il(ε_{m}) for m = 0, …, 4]

2.3. Global coordinates. Let I[j, i) = [⟨jτ⟩, ⟨iτ⟩) be an interval from I = [0, 1) for any integers j, i. We assume that ⟨jτ⟩ < ⟨iτ⟩, where ⟨ · ⟩ is the fractional part. If i = 0, we define I[j, 0) = [⟨jτ⟩, 1⟩).

Let m be even. Then eigen Fibonacci tilings T il(ε_{m}) (2.3) are

T il(ε_{m}) = ∑_{0≤k<F_{m+1}}(G_{k}^{m} + E_{k}^{m}) + ∑_{F_{m+1}≤k<F_{m+2}} E_{k}^{m},  (2.10)

split into intervals G_{k}^{m} and E_{k}^{m} determined by the equalities

G_{k}^{m} = I[k^{+}, k^{−}), E_{k}^{m} = I[k^{−}, k).  (2.11)

Here we use the notation k^{+} = k + F_{m}, k^{−} = k − F_{m+1}. The intervals G_{k}^{m}, E_{k}^{m} from tiling (2.10) have lengths (2.6). The mapping S connects G_{k}^{m}, E_{k}^{m} and the initial intervals G_{0}^{m}, E_{0}^{m} by the shift

G_{k+1}^{m} ≡ G_{k}^{m} + τ mod 1,  (2.12)

where 0 ≤ k < F_{m+1} and G_{0}^{m} = I[F_{m}, −F_{m+1}) = I[⟨F_{m}⟩, ⟨−F_{m+1}⟩), and

E_{k+1}^{m} ≡ E_{k}^{m} + τ mod 1,  (2.13)

where 0 ≤ k < F_{m+2} and E_{0}^{m} = I[−F_{m+1}, 0) = [⟨−F_{m+1}τ⟩, 1).

Similarly, for odd levels m eigen tilings T il(ε_{m}) (2.3) are represented as the sum

T il(ε_{m}) = ∑_{0≤k<F_{m+1}}(G_{k}^{m} + E_{k}^{m}) + ∑_{F_{m+1}≤k<F_{m+2}} G_{k}^{m},  (2.14)

In this case the intervals G_{k}^{m} and E_{k}^{m} are equal to

G_{k}^{m} = I[k^{−}, k^{=}), E_{k}^{m} = I[k^{=}, k),  (2.15)

where k^{−} = k − F_{m}, k^{=} = k − F_{m+2}. They are connected by the S-shift

G_{k+1}^{m} ≡ G_{k}^{m} + τ mod 1,  (2.16)

where 0 ≤ k < F_{m+2}, G_{0}^{m} = I[−F_{m}, −F_{m+2}) = [⟨−F_{m}τ⟩, ⟨−F_{m+2}τ⟩), and

E_{k+1}^{m} ≡ E_{k}^{m} + τ mod 1,  (2.17)

where 0 ≤ k < F_{m+1} and E_{0}^{m} = I[−F_{m+2}, 0) = [⟨−F_{m+2}τ⟩, 1).

Formulas (2.11) and (2.15) define global coordinates for the intervals G_{k}^{m} and E_{k}^{m} in the eigen Fibonacci tilings T il(ε_{m}).

3. Fibonacci numbers and the golden section

3.1. Periodicity of Fibonacci numbers mod a. Let us consider a sequence (see [7])

F_{1}, F_{2}..., F_{m}, ... mod a  (3.1)

of the Fibonacci numbers determined by the recurrence formula

F_{m+2} = F_{m} + F_{m+1}  (3.2)

for m ≥ 1 and the initial conditions F_{1} = 1, F_{2} = 1. Let’s find in the sequence (3.1) the first repeating pair of the Fibonacci numbers

F_{m+T} ≡ F_{m} mod a, F_{m+1+T} ≡ F_{m+1} mod a.  (3.3)

Let T = T (a) be the least number for which congruences (3.3) holds for some index m. By m_{1} = m_{1}(a) we denote the least of such m. Since the number of all integer pairs moda is equal to a^{2}, therefore the function T (a) is well defined and we have the trivial inequality

T (a) ≤ a^{2}.

Subtracting in (3.3) the first congruence from the second congruence and using the recurrence formula for Fibonacci numbers, we get one more pair of the congruences

F_{m−1+T} ≡ F_{m−1} mod a, F_{m+T} ≡ F_{m} mod a.

It means that in congruences (3.3) the index m can be moved on one step to the left, if m ≥ 2. Similarly, by addition of congruences (3.3), we see that the index m also can be moved to the right. Therefore T is the period of sequence (3.1) and

m_{1} = 1 for any a ≥ 1.

Hence, in sequence (3.1) the first repeating pair is

F_{T +1} ≡ 1 mod a, F_{T +2} ≡ 1 mod a.  (3.4)

From (3.4) we obtain the congruence F_{T} ≡ 0 mod a. Suppose the congruence

F_{t} ≡ 0 mod a

holds for some t. If we assume a > 1, then using the recurrence formula for Fibonacci numbers we have

F_{t+1} ≡ f mod a, F_{t+2} ≡ f mod a,  (3.5)

and f ≡ F_{t−1} mod a. Let t(a) be a least t so that a divides the Fibonacci number F_{t}. The number t is called a period of self-similarity of sequence (3.1) and f is a coefficient of self-similarity of the sequence. Note that the functions t(a) and T (a) have the following multiplicative property

t(a_{1} · a_{2}) = l.c.m.(t(a_{1}), t(a_{2})), T (a_{1} · a_{2}) = l.c.m.(T (a_{1}), T (a_{2})),  (3.6)

if a_{1} and a_{2} are coprime numbers.

We can write congruences (3.5) as

F_{t+1} ≡ fF_{1} mod a, F_{t+2} ≡ fF_{2} mod a.

Using these congruences and the recurrence formula, we obtain the self-similarity property of sequence (3.1):

F_{m+t} ≡ fF_{m} mod a

for all m = 1, 2, 3... Substituting t + 1, t + 2, and 2t + 1, 2t + 2, etc. for m in the last congruence and using (3.5), we get the congruences

F_{it+1} ≡ f^{i} mod a, F_{it+2} ≡ f^{i} mod a,  (3.7)

where i = 1, 2, 3... and f = f(a) is the coefficient of self-similarity of sequence (3.1).

Suppose congruence (3.5) holds with some f. We assert that a and f are coprime numbers. If the greatest common divisor of the numbers a and f is equal to d, then from congruence (3.5) follows that F_{t+2} ≡ 0 mod d, F_{t+1} ≡ 0 mod d,…, F_{1} ≡ 0 mod d, i.e. d = 1.

Since the numbers a and f are coprime, there exists an exponent δ > 0 such that

f^{δ} ≡ 1 mod a.

Let δ = δ(a) be the order f mod a. It is easy to see, that the order δ can accept only the values 1, 2 or 4. Really, from the formula F_{t−1}^{2} = F_{t−2}F_{t} + (−1)^{t} and the condition a|F_{t} we get the congruence

F_{t−1}^{2} ≡ (−1)^{t} mod a.

The coefficient of self-similarity satisfies the congruence f ≡ F_{t−1} mod a. By the above two congruence, we get the following statements:

δ = 1 or 2, if t is even ,
δ = 4,  if t is odd and a ≥ 3.

(3.8)

Besides, the order δ = 1 for a = 1 or 2.

Let us prove that the periods T = T (a), t = t(a) of sequence (3.1) and the order δ = δ(a) are connected by the formula

T (a) = t(a)δ(a)  (3.9)

for all a = 1, 2, 3... First, we notice that t(a) ≤ T (a). Let’s divide T (a) = kt(a) + r, where r is a remainder 0 ≤ r < t(a). Then by congruences (3.7) and (3.6) we can write

F_{kt+1} ≡ F_{kt+2} mod a, F_{T +1} ≡ F_{T +2} mod a.

From the first congruence it follows that the module a is coprime with the Fibonacci numbers F_{kt+1} and F_{kt+2}. Therefore

F_{r+1} ≡ F_{r+2} ≡ f mod a, where f ≡ F_{T +1}F_{kt+1}^{−1} mod a.

Hence, F_{r} ≡ 0 mod a. As 0 ≤ r < t(a), from the definition of the period t(a) we obtain r = 0.

It is known that the period T (a) for a prime module a = p has properties:

T (p) divides p − 1, if p ≡ 1 or 4 mod 5,

T (p) divides 2(p + 1), if p ≡ 2 or 3 mod 5.

3.2. Fibonacci numbers and Euler function. Consider the real quadratic field ℚ(\sqrt{5}). Its ring of integers ℤ[τ] has the basis 1, τ = (1 + \sqrt{5})/2 over the ring ℤ. It is an Euclidian ring and τ is a unit of the given ring, because τ^{−1} = τ − 1. Under the Euler theorem in the ring ℤ[τ] for each a = 1, 2, 3... it holds the congruence

τ^{Φ(a)} ≡ 1 mod aℤ[τ],  (3.10)

where Φ is the Euler function of the ring ℤ[τ]. The restriction Φ onto the set of natural numbers ℕ = {1, 2, 3, …} represents a multiplicative function with the following values on degrees of prime numbers:

Φ(p^{α}) = p^{2α−2}Φ(p)  (3.11)

for prime numbers p ∈ ℕ and α ≥ 1, and

Φ(p) = (p − 1)^{2}, if p ≡ 1 or 4 (mod 5),
Φ(p) = p^{2} − 1, if p ≡ 2 or 3 (mod 5),
Φ(p) = p(p − 1), if p = 5.

Let Δ = Δ(a) be an order of τ in the ring ℤ[τ], i.e. the least Δ > 0, Δ ∈ ℤ with the condition

τ^{Δ} ≡ 1 mod aℤ[τ].  (3.12)

Let’s take the well-known formula

τ^{Δ} = F_{Δ−1} + F_{Δ}τ

connecting the golden section τ with the Fibonacci numbers. Comparing this formula with condition (3.12), we get the congruences

F_{Δ−1} ≡ 1 mod a, F_{Δ} ≡ 0 mod a

in the ring ℤ. Conversely, from the last congruences it follows congruence (3.12). Besides, combining the last congruences with the recurrence formula for Fibonacci numbers(3.2) we have two congruences

F_{Δ+1} ≡ 1 mod a, F_{Δ+2} ≡ 1 mod a.

Taking into account minimality of the order Δ(a) (3.12) we obtain that the order Δ(a) is equal to the period T (a) of sequence (3.1).

Proposition 1. 1. Suppose a is any natural number, T (a) is the period of the sequence of Fibonacci numbers F_{1}, F_{2}..., F_{m}, ... mod a, and t(a) is its period of selfsimilarity. Then the following equality is true

T (a) = Δ(a),

where Δ(a) is the order of τ mod aℤ[τ]. The periods T (a) and t(a) divide Φ(a), where Φ is the Euler function of the ring ℤ[τ]. The multiplicative function Φ is determined by equalities (3.11).

2. For m ≥ 4 it follows that

T (F_{m}) = 2m, if m even,
T (F_{m}) = 4m, if m odd. (3.13)

The equalities T (F_{1}) = T (F_{2}) = 1 and T (F_{3}) = 3 hold for indexes m < 4. In particular, from (3.13) it follows that 2m or 4m divides Φ(F_{m}), if m ≥ 4 and m is even or odd respectively.

Proof. 1. The equality T (a) = Δ(a) is already proved. The fact that T (a) and t(a) divide Φ(a) follows from (3.10), (3.13), and (3.9).

2. Let a = F_{m}. In this case the period of self-similarity t is equal to

t(F_{m}) = m  (3.14)

for indexes m ≥ 3, and t(F_{1}) = t(F_{2}) = 1. The coefficient of self-similarity f (3.5) satisfies the congruence

f ≡ F_{m−1} mod F_{m}.

Taking into account that the congruence F_{m−1} ≡ 1 mod F_{m} holds only for indexes m = 1, 2, 3, formulas (3.8) take the form for the module a = F_{m}

δ(F_{m}) = 2 for even m ≥ 4,
δ(F_{m}) = 4 for odd m ≥ 4, (3.15)

and δ(F_{m}) = 1 for the indexes m = 1, 2, 3. Finally, combining (3.14) and (3.15) we transform formula (3.9) to formulas (3.13). □

In Table 3.1 for sequence of Fibonacci numbers mod a (3.1) we give the values of the periods T = T (a), the periods of self-similarity t = t(a), the coefficients of self-similarity f = f(a) ≡ F_{t−1} mod a, 0 < f ≤ a, the orders δ = δ(a) of the coefficients f mod a, and the Euler function Φ of the ring ℤ[τ], where 1 ≤ a ≤ 21.

\begin{matrix}a & t & f & δ & T & Φ \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 2 & 3 & 1 & 1 & 3 & 3 \\ 3 & 4 & 2 & 2 & 8 & 8 \\ 4 & 6 & 1 & 1 & 6 & 12 \\ 5 & 5 & 3 & 4 & 20 & 20 \\ 6 & 12 & 5 & 2 & 24 & 24 \\ 7 & 8 & 6 & 2 & 16 & 48 \\ 8 & 6 & 5 & 2 & 12 & 48 \\ 9 & 12 & 8 & 2 & 24 & 72 \\ 10 & 15 & 7 & 4 & 60 & 60 \end{matrix}  \begin{matrix}a & t & f & δ & T & Φ \\ 11 & 10 & 1 & 1 & 10 & 100 \\ 12 & 12 & 5 & 2 & 24 & 96 \\ 13 & 7 & 8 & 4 & 28 & 168 \\ 14 & 24 & 13 & 2 & 48 & 144 \\ 15 & 20 & 11 & 2 & 40 & 160 \\ 16 & 12 & 9 & 2 & 24 & 192 \\ 17 & 9 & 4 & 4 & 36 & 288 \\ 18 & 12 & 11 & 2 & 24 & 216 \\ 19 & 18 & 1 & 1 & 18 & 324 \\ 20 & 30 & 9 & 2 & 60 & 240 \\ 21 & 8 & 13 & 2 & 16 & 384 \end{matrix}

Table 3.1.

3.3. Distribution of two points on a circle. We identify the unit interval I = [0, 1) with the unit circle ℝ/ℤ and consider the simple shift

S(x) = x + τ mod 1,

where τ = (1 + \sqrt{5})/2 is the golden section. The unit interval is tiled

T il(ε_{0}) = G_{0}^{0} ⊕ E_{0}^{0}

into two intervals G_{0}^{0} = [0, τ^{−2}) and E_{0}^{0} = [τ^{−2}, 1). By the sign ⊕ we denote a non-commutative operation of applying intervals. For any point x ∈ I we consider its orbit {x_{i}}_{i=0}^{∞}, where x_{i} = S^{i}(x). Define the function ε(x_{i}) (the indicator of G_{0}^{0}) by the condition

ε(x_{i}) = \begin{cases}1, & if x ∈ G_{0}^{0}, \\ 0, & if x ∈ E_{0}^{0}. \end{cases}

Let s_{k} = ε(x_{1}) + ⋯ + ε(x_{k}) for k = 1, 2, 3... and

s_{k} = max s_{k}(x), x ∈ I.  (3.16)

As the fractional part ⟨τ⟩ is greater than the length τ^{−2} of the interval G_{0}^{0}, in the sequence ε(x_{1}), …, ε(x_{k}) two units can not be in succession. From here it follows the trivial upper estimation for the function s_{k}

s_{k} ≤ [\frac{k + 1}{2}],

where the square brackets denote the integer part of a number.

To prove the exact formula for the function s_{k} (see (3.16)) we transform the tiling T il(ε_{0}) to the second level tiling

T il(ε_{2}) = E_{2}^{2} ⊕ G_{1}^{2} ⊕ E_{1}^{2} ⊕ G_{0}^{2} ⊕ E_{0}^{2},

where G_{0}^{0} = E_{0}^{2} ⊕ G_{1}^{2}, and interval E_{0}^{0} = E_{1}^{2} ⊕ G^{2} ⊕ E_{0}^{2} is tiled symmetrically. All possible pairs of points (x, y) ∈ I × I we split

I × I = M_{1} + ⋯ + M_{5}

into subsets M_{i}, i = 1, …, 5, consisting of pairs (x, y) for which the fractional part ⟨y − x⟩ belongs to i-th interval of the tiling T il(ε_{2}). The intervals are numbered from left to right. The pair (ε(x) ε(y)) is called the state of the pair (x, y). The following graphs specify an admissible S-transitions

S : (ε(x), ε(y)) ↦ (ε(S(x)), ε(S(y)))

of a pair (x, y) from one state to another.

(1 1)  (1 1)
↑↓  ↓↑
(0 0)_{⟳}  (0 0)_{⟳}
↙  ↖  ↗  ↖
(1 0) ⟶ (0 1)  (1 0) ⟵ (0 1)
(M_{1})  (M_{4})
(0 0)_{0}  (0 0)_{1}
↓ ↗ ↖ ↓
(1 0)_{0} ⇄ (0 1)_{1}
(M_{3})
(1 1)  (1 1)
↑↓  ↓↑
(0 0)_{⟳}  (0 0)_{⟳}
↙  ↖  ↗  ↖
(1 0) ⇄ (0 1)  (1 0) ⇄ (0 1)
(M_{2})  (M_{5})

Fig. 3.1. The graphs (M_{i}), i = 1, …, 5 of the admissible S-transitions from one state to another

In the graph (M_{3}) the state (0 0) splits into the states (0 0)_{0} and (0 0)_{1}. A pair (x, y) is found in the first state, if x ∈ E_{1}^{2}, and it is found in the second state, if y ∈ E_{1}^{2}.

For any y ∈ I we define the difference

Δ_{k}(0, y) = s_{k}(0) − s_{k}(y),

where k = 1, 2, 3.... We assume (0, y) ∈ M_{1}. Then the pair (0, y) has the initial state (1 1). We act on (0, y) by the shift S:

(0, y) → (S(0), S(y)) → ⋯ → (S^{k}(0), S^{k}(y)).

By the graph (M_{1}), under the shift S the pair (0, y) changes its state according to the following chain

(1 1)_{=} → ⋯_{=} → (1 0)_{↗} → (0 1)_{=} → ⋯_{=}

Here the signs = and ↗ show that on the given step the difference is equal to 0 or 1 respectively, i.e. Δ_{k}(0, y) = 1 for k with the state (1 0) and △_{k}(0, y) = 0 in other cases. Thus, the difference is equal to

Δ_{k}(0, y) = 0 or 1  (3.17)

for all k = 1, 2, 3..., if pair (0, y) ∈ M_{1}. In the case, when the pair (0, y) belongs to other subsets M_{i}, i = 2..., 5, equality (3.17) can be proved similarly by the help of S-transitions graph (M_{i}).

From the definition of the function s_{k} (see (3.16) and (3.17)) follows the equality

s_{k} = s_{k}(0)  (3.18)

for k = 1, 2, 3, . . .. Given equality specifies that the extremal function s_{k} admits the global realization at x = 0.

Let’s find the exact formula for the sum s_{k}(0). Let R(k) be a number of points of the orbit {0_{i}}_{i=0}^{∞} with numbers 1 ≤ i ≤ k that visit the interval E_{0}^{0}. The following formula for R(k) is proved in Lemma 6.13 [10]

R(k) = [kτ^{−1}].  (3.19)

The unit interval I is split into the intervals G_{0}^{0} and E_{0}^{0}. We can write the equality s_{k}(0) = k − R(k). Using (3.19) we get s_{k}(0) = k − [kτ^{−1}]. Notice that [−kτ^{−1}] = −[kτ^{−1}] − 1 and 1 − τ^{−1} = τ^{−2}. Thus we have the required formula

s_{k}(0) = [kτ^{−2}] + 1  (3.20)

for k = 1, 2, 3...

Proposition 2. For extremal function s_{k} (3.16) the following exact formula takes place

s_{k} = [kτ^{−2}] + 1  (3.21)

for k = 1, 2, 3...

The proof follows from equality (3.18) and formula (3.20). □

4. Boundary derivatives and self-similarity

4.1. Derivatives of the simple shift. We write the Fibonacci tiling T il(ε_{m}) (see (2.10), (2.11)) as the sum

T il(ε_{m}) = X ⊕ G_{0}^{m} ⊕ E_{0}^{m}.  (4.1)

Suppose the tiling T il(ε_{m}) has an even level m. Then the simple shift S : x ↦ x + τ mod 1 of the unit interval I = [0, 1) translates T il(ε_{m}) in the tiling

S(T il(ε_{m})) = X ⊕ E_{F_{m+2}}^{m} ⊕ G_{F_{m+1}}^{m}

Therefore the action of the shift S on the tiling T il(ε_{m}) is reduced to a rearrangement of two right intervals

G_{0}^{m} ⊕ E_{0}^{m} = E_{F_{m+2}}^{m} ⊕ G_{F_{m+1}}^{m}.  (4.2)

Denote by I^{m} the sum G_{0}^{m} ⊕ E_{0}^{m} . Let e_{m} = |E_{0}^{m}| be the length of the interval E_{0}^{m}. By formulas (2.6), (2.7), e_{m} = τ^{−(m+1)} and

|I^{m}| = τ^{−m}  (4.3)

for any level m. Therefore we have

\frac{e_{m}}{|I^{m}|} = τ^{−1} = τ − 1 ≡ τ mod 1.

Hence for even m the derivative S|I^{m} , i.e. restriction of the shift S on the intervals I^{m}, is similar to the shift S:

S|I^{m} ∼ S.  (4.4)

Similarly, if the level m is odd, using formulas (2.14) and (2.15) we obtain

S(T il(ε_{m})) = X ⊕ E_{F_{m+1}}^{m} ⊕ G_{F_{m+2}}^{m}.

The action of the shift S, by 4.1, is reduced to the rearrangement of intervals

G_{0}^{m} ⊕ E_{0}^{m} = E_{F_{m+1}}^{m} ⊕ G_{F_{m+2}}^{m}.  (4.5)

By the formulaes (2.7) and (4.3), for odd m we have

\frac{e_{m}}{|I^{m}|} = τ^{−2} = 2 − τ ≡ −τ mod 1.

Thus the derivative

S|I^{m} ∼ S^{−1}  (4.6)

is similar to the inverse shift S^{−1} : x ↦ x − τ mod 1 of the unit interval I.

4.2. Upper derivatives of two-colour shifts. For any a and b from the set ℤ = {0, ±1, ±2...}, ab > 0, we define the two-colour or the composite shift S_{ε}(a, b) of the unit interval I by the conditions

S_{ε}(a, b)(x) = x + aτ mod 1, if x ∈ I_{ε}^{−},
S_{ε}(a, b)(x) = x + bτ mod 1, if x ∈ I_{ε}^{+}. (4.7)

Here ε is a continuous parameter of the composite shift S_{ε}(a, b), that ranges over the unit interval I. Define the colouring of the interval I by the tiling

I = I_{ε}^{−} ⊕ I_{ε}^{+}

into the intervals I_{ε}^{−} = [0, ε) and I_{ε}^{+} = [ε, 1). The aim of this section is to consider the upper derivatives

S_{ε}(a, b)^{(m)} = S_{ε}(a, b)|I^{m} × I^{m}  (4.8)

of composite shift (4.8) for any m = 0, 1, 2.... The upper derivatives are a restriction of the shift S_{ε}(a, b) for x and ε both on the interval I^{m} = [ε_{m}, 1). Here ε_{m} = 1−τ^{−m} are eigenvalues (2.2) of the B-process.

First, we consider the composite shift S_{ε}(a, b) with the parameters a, b from the set ℕ = {1, 2, 3, …}. Write b as

b = qa + b_{1},

where 1 ≤ b_{1} ≤ a. For any point x ∈ I^{m} denote by Orb(x) its S-orbit. It consists of the elements x_{i} = S^{i}(x) ≡ x + iτ mod 1. Here i = 0, 1, 2... is the S-number of the points x_{i} from the orbit Orb(x). Similarly, consider the S_{ε}(a, b)-orbit Orb_{ε}(x) which consists of the elements x_{j′} = S_{ε}^{j}(a, b)(x), j = 0, 1, 2.... Here j is the S_{ε}(a, b)- number of the element x_{j′} with the S-number that is equal to j′.

Suppose the point x belongs to the interval I_{+}^{m} = I^{m} ∩ I_{ε}^{+} from I^{m}. Then by definition (4.7) the orbit Orb_{ε}(x)\{x} consists of elements with the S-numbers

b, b + a, …, b + ka, …  (4.9)

These numbers represent the whole progression i ≡ b_{1} mod a beginning from i = b_{1}. We consider all the elements of the orbit Orb_{ε}(x)\{x} up to the first returning of the point x in the interval I^{m}. Further we impose additional restrictions on the interval I^{m} which ensure the returning of x in I^{m}. If the point x belongs to the interval I_{−}^{m} = I^{m} ∩ I_{ε}^{−} from I^{m}, the elements of its orbit Orb_{ε}(x)\{x} have the S-numbers

a, 2a, …, ka, …  (4.10)

forming the progression i ≡ 0 mod a beginning from i = a.

According to Subsection 2.3 and equalities (4.2), (4.3), for any point x from the interval I^{m} = G_{0}^{m} ⊕ E_{0}^{m} the minimal S-number i > 0 of elements x_{i} from Orb(x) ∩ I^{m} is equal to

\begin{matrix}i = F_{m+1} & if x ∈ G_{0}^{m}, \\ i = F_{m+2} & if x ∈ E_{0}^{m}, \end{matrix}  (4.11)

for even levels m. Similarly,

\begin{matrix}i = F_{m+2} & if x ∈ G_{0}^{m}, \\ i = F_{m+1} & if x ∈ E_{0}^{m}, \end{matrix}

for odd m.

Now we need to connect the Fibonacci numbers F_{m+1} and F_{m+2} with the shift’s parameter a. First, we assume

F_{m+1} ≡ F_{m+2} ≡ 1 mod a.

Let x_{i} be elements consistently returned by the shift S in the interval I^{m}. Then their S-numbers are equal to

j_{1}, j_{2}, …, j_{k}, …

Here j_{k} = i_{1} + i_{2} + ⋯ + i_{k} is the number of the element x_{j_{k}} ∈ I^{m} corresponding to the k-th returning of the point x in the interval I^{m}.

Let’s calculate the least numbers k for which j_{k} gets into the progression b + aℕ (see (4.9)) or the progression aℕ (see (4.10)). From (4.11) it follows that any number i_{l} ≡ 1 mod a. Hence

j_{k} ≡ k mod a.

We estimate the values of j_{k} with the help of the function s_{k} (see (3.16)). If the level m is even, then using (4.4) and (4.11) we obtain the inequality

j_{k} = i_{1} + i_{2} + ⋯ + i_{k} ≥ s_{k}F_{m+1} + (k − s_{k})F_{m+2}.

Using the notation s(k, m) = kF_{m+2} − s_{k}F_{m}, we can write this inequality as

j_{k} ≥ s(k, m),  (4.12)

where k = 1, 2, 3... For odd m we use formula (4.6) and the function s_{k} and replace the shift S by its inverse shift S^{−1}. Repeating the above proof we return to inequality (4.12). For the inverse function s_{k} we again use exact formula (3.21).

Let’s choose k = b_{1}. The number j_{k} belongs to the progression b + aℕ if and only if the inequality j_{k} ≥ b is true. Assume that b ≤ s(b_{1}, m). Using (4.12) we see that this inequality takes place. So, for any point x ∈ I_{+}^{m} the first element x_{j′} from the S_{ε}(a, b)-orbit Orb_{ε}(x), returning to the interval I^{m}, has the S-number

j′ = j_{b_{1}} if b ≤ s(b_{1}, m).  (4.13)

Further we choose k = a. Then the number j_{k} belongs to the progression aℕ as j_{k} ≡ 0 mod a. By (4.11), j_{k} = i_{1} + i_{2} + ⋯ + i_{k} ≥ a. Suppose the initial point x belongs to the interval I_{−}^{m} and x_{j′} is the first element from the orbit Orb_{ε}(x) returning to the interval I^{m}. Then we have

j′ = j_{a}.  (4.14)

Let’s consider more general restrictions for the Fibonacci numbers

F_{m+1} ≡ F_{m+2} ≡ F mod a,

where 1 ≤ F ≤ a. In the Subsection 3.1 we proved that the coprimity of the numbers F and a follows from these congruences. If F = a, then a = 1. We denote by (bF^{−1})_{a} a least positive solution x ≥ 1 of the congruence xF ≡ b mod a. So, if F = 1, then (bF^{−1})_{a} is equal to to the remainder 1 ≤ b_{1} ≤ a as before. For k = (bF^{−1})_{a} we have the congruence

j_{k} ≡ b_{1} mod a.

Hence the S-number j_{k} belongs to the progression b + aℕ, if b ≤ s((bF^{−1})_{a}, m). Let x_{j′} be the first element from the S_{ε}(a, b)-orbit Orb_{ε}(x) returning to the interval I^{m}. Here the initial point x belongs to I_{+}^{m}. Then x_{j′} has the number

j′ = j_{(bF^{−1})_{a}} if b ≤ s((bF^{−1})_{a}, m).  (4.15)

Under the condition F = 1 formula (4.15) transforms to formula (4.13). If we choose an initial point x from the interval I_{−}^{m}, the S-number j′ of the first point x_{j′} returning to the interval I^{m} again satisfies equality (4.14).

Finally we consider the composite shift S_{ε}(a, b) with integer negative parameters a, b. For the negative parameters a, b it is possible to interpret the shift S_{ε}(a, b) as composite shift (4.7) with the positive parameters −a, −b replacing τ by −τ. Thus it is necessary to replace the simple shift S from Subsection 4.1 by its inverse shift S^{−1} and to change the order of rearrangement (4.2) and (4.5). After the specified changes formula (4.15) takes the form

j′ = j_{(|b|F^{−1})_{a}} if |b| ≤ s((|b|F^{−1})_{a}, m).  (4.16)

Here the coefficient F satisfies the inequalities 1 ≤ F ≤ |a|. Formula (4.14) transforms to the form

j′ = j_{|a|}.

Theorem 1. Let S_{ε}(a, b) be a composite shift (see (4.7)) of the unit interval I = [0, 1) with the integer parameters a, b so that ab > 0. Suppose for the index m = 0, 1, 2... the Fibonacci numbers F_{m+1}, F_{m+2} satisfy the congruences

F_{m+1} ≡ F_{m+2} ≡ F mod a,  (4.17)

where 1 ≤ F ≤ |a|. Suppose |b| ≤ s((bF^{−1})_{a}, m) for the function s(k, m) from (4.12). Then for the upper derivatives of the composite shift S_{ε}(a, b)^{(m)} (see (4.8)) the following statements hold:

S_{ε}(a, b)^{(m)} ∼ S_{ε′}(a, sign(b) · b′),  (4.18)

if m is even. Here

b′ = (|b|F^{−1})_{a}, ε′ = (ε − ε_{m})/(1 − ε_{m})  (4.19)

are renormalisations of the parameters b, ε, sign(b) = ±1 is the sign of b, and ε_{m} = 1 − τ^{−m} is the eigenvalues of the B-process (see (2.2));

S_{ε}(a, b)^{(m)} ∼ S_{ε′}(−a, −sign(b) · b′),  (4.20)

if m is odd, ε′ and b′ are the same renormalisations of parameters as in (4.19). The similarity of the composite shifts in (4.18) and (4.20) means their conjugacy

S_{ε}(a, b)^{(m)} = h_{m} ∘ S_{ε′}((−1)^{m}a, (−1)^{m}sign(b) · b′) ∘ h_{m}^{−1}

under the affine isomorphism h_{m}(x) = τ^{−m}(x − 1) + 1 of the unit interval I and the differentiation interval I^{m} = [ε_{m}, 1).

Proof. Using tilings (4.2) and (4.5) we see that each returning of an initial point x ∈ I^{m} to the interval I^{m} corresponds to rearrangement of the intervals G_{0}^{m} and E_{0}^{m}. The rearrangement is equivalent to the simple shift S or S^{−1} of the unit interval, if the level m is even or odd, respectively. The S-numbers of the first returnings of the initial point x to the interval I^{m} are calculated by formulas (4.13) - (4.16). Thus, using the definition of the composite shift S_{ε}(a, b) (see (4.7)), we obtain similarity formulas (4.18) and (4.20), and the renormalisation b′ of parameter b. Renormalisation ε′ (4.19) of the parameter ε and the affine isomorphism h_{m} follow from the choice of the differentiation interval I^{m} = [ε_{m}, 1) and from the equality ε_{m} = 1 − τ^{−m}.

□

4.3. Self-similarity of the composite shift. For the composite shift S_{ε}(a, b) we consider the upper derivatives S_{ε}(a, b)^{(m)} of the order m = kt, where k = 0, 1, 2... and t = t(a) is equal to the self-similarity period of the sequence of the Fibonacci numbers mod a (see (3.1)). By theorem 1, for m = kt we have

S_{ε}(a, b)^{(kt)} ∼ S_{ε′}((−1)^{kt}a, (−1)^{kt}sign(b) · b^{(k)})

with the renormalization b^{(k)} = (|b|f^{−1})_{a} and the self-similarity coefficient f = f(a) (see (3.5)) of the sequence of the Fibonacci numbers mod a. In particular, for k = δ, where δ = δ(a) is the order of f, we obtain b^{(δ)} = (|b|)_{a}. If we impose the restriction 1 ≤ |b| ≤ |a| for the parameter b, we can write sign(b) · b^{(δ)} = b. Hence,

S_{ε}(a, b)^{(T)} ∼ S_{ε′}((−1)^{T} a, (−1)^{T} b).

Here T = δt is the period of the sequence of the Fibonacci numbers mod a. In the case, when the period T is even, it is follows the property of self-similarity of the composite shift

S_{ε}(a, b)^{(T)} ∼ S_{ε′}(a, b),

i.e. the restriction of S_{ε}(a, b) onto the square I^{T} × I^{T} (4.8) is similar to the shift S_{ε}(a, b). If the period T is odd, we choose k = 2δ and again come to the selfsimilarity

S_{ε}(a, b)^{(2T)} ∼ S_{ε′}(a, b).

So, we have proved

Corollary 1. The composite shift S_{ε}(a, b) (see (4.7)) with the integer parameters a, b so that ab > 0 and 1 ≤ |b| ≤ |a| has the following self-similarity property

S_{ε}(a, b)^{(m)} ∼ S_{ε′}(a, b).  (4.21)

The renormalization ε′ of the parameter ε is calculated by formula (4.19). The derivative order m in (4.21) is equal to T or 2T, if the period T = T (a) of the sequences of the Fibonacci numbers mod a is even or odd, respectively. □

If the parameter a of the composite shift S_{ε}(a, b) is equal to a prime number p ≢ 0 mod 5, we have a simple connection between p and the period T = T (p) of the sequence of the Fibonacci numbers mod p (cf. (3.1)). For p = 5 the value of the period T (5) is specified in Table 3.1.

Corollary 2. The upper derivatives S_{ε}(p, b)^{(p)} of the prime order p for the composite shift S_{ε}(p, b) (see (4.7)) with the parameters 1 ≤ b ≤ p have the following self-similarity properties :

S_{ε}(p, b)^{(p−1)} ∼ S_{ε′}(p, b)

for the prime numbers p ≡ 1 or 4 mod 5,

S_{ε}(p, b)^{(2(p+1))} ∼ S_{ε′}(p, b)

for the prime numbers p ≡ 2 or 3 mod 5, and

S_{ε}(p, b)^{(20)} ∼ S_{ε′}(p, b)

for p = 5. □

We know the exact formula for the period T = T (a) of sequence (3.1) only for a = F_{m}. By formulas (3.13), for m ≥ 4 the period T (F_{m}) is equal to 2m if m is even, and it is equal to 4m if m is odd. In other cases the period T (F_{m}) is equal to 1 for m = 1, 2 and, by Table 3.1, T (F_{m}) is equal to 3 for m = 3. Thus, from formula (4.21) it follows

Corollary 3. If m ≥ 4 and 1 ≤ b ≤ F_{m}, the following self-similarity properties of the composite shifts S_{ε}(F_{m}, b) take place:

S_{ε}(F_{m}, b)^{(2m)} ∼ S_{ε′}(F_{m}, b)

if m is even, and

S_{ε}(F_{m}, b)^{(4m)} ∼ S_{ε′}(F_{m}, b)

if m is odd. For m = 2 the self-similarity formula

S_{ε}(1, 1)^{(2)} ∼ S_{ε′}(1, 1)

is equivalent to the formula S|I^{2} ∼ S (see (4.4)) for the simple shift S : x ↦ x + τ mod 1. For m = 3 we have the formula

S_{ε}(2, 1)^{(6)} ∼ S_{ε′}(2, 1).
□

Comparing Corollary 2 and Corollary 3, we see that the composite shifts S_{ε}(a, b) with the parameter a = F_{m} have very short periods of the self-similarity. We notice that δ(F_{m}) = 2 (3.15), f = F_{m−1}, F_{m−1}^{2} ≡ 1 mod F_{m} for even m ≥ 4 and δ(F_{m}) = 4, F_{m−1}^{2} ≡ −1 mod F_{m} for odd m. Therefore, by theorem 1 we get the following formulas for the upper derivatives S_{ε}(a, b)^{(m)} of the composite shift S_{ε}(a, b) inside the self-similarity period:

S_{ε}(F_{m}, F_{m−1})^{(2m)} ∼ S_{ε′}(F_{m}, F_{m−2}),  (4.22)
S_{ε}(F_{m}, F_{m−2})^{(2m)} ∼ S_{ε′}(F_{m}, F_{m−1})

for even m ≥ 4, and

S_{ε}(F_{m}, F_{m−1})^{(m)} ∼ S_{ε′}(F_{m}, 1),
S_{ε}(F_{m}, 1)^{(m)} ∼ S_{ε′}(F_{m}, F_{m−1})

for odd m ≥ 4.

Generally, the composite shift S_{ε}(a, b) is a contracting mapping

I ⊃ S_{ε}(a, b)(I) ⊃ … ⊃ S_{ε}^{k}(a, b)(I) ⊃ …

For any integer parameters a, b, ab > 0, this chain contains only a finite number of strict inclusions. The attractor of the composite shift S_{ε}(a, b) is the set

Att(S_{ε}(a, b)) = ⋂_{k=0}^{∞} S_{ε}^{k}(a, b)(I),

where we denote by S_{ε}^{k}(a, b) the k-multiple composition of the shift S_{ε}(a, b). In Theorem 1 let us choose the parameters a, b ∈ ℤ, ab > 0, so that a divides b. Assume that in formulaes (4.18) and (4.20) the order of upper derivatives m is divided into the self-similarity period t = t(a) of sequence (3.1) of the Fibonacci numbers . Then we have the similarity

S_{ε}(a, b)^{(m)} ∼ S^{a}  (4.23)

for even m, and

S_{ε}(a, b)^{(m)} ∼ S^{−a}

for odd m. Here S is the simple shift of the unit interval I = [0, 1) from Subsection 3.3. By the above definition of attractor, the equality Att(S^{a}) = I takes place for all a ≠ 0. Since τ is irrational, the shift S^{a} : x ↦ x + aτ mod 1 has a minimality property, i.e. the orbit of any point x ∈ I is dense in I. In other words, the shift S^{a} has not closed invariant proper subsets in I. Recall that, by definition (4.8), the upper derivative of the composite shift S_{ε}(a, b)^{(m)} is a restriction of the shift S_{ε}(a, b) and its parameter ε onto the interval I^{m} = [1 − τ^{−m}, 1). Then from (4.23) it follows

Corollary 4. Suppose the composite shift S_{ε}(a, b) (see (4.7)) has the parameters a, b of the same sign, a divides b, and m is divided by the self-similarity order t = t(a) of sequence (3.1) of the Fibonacci numbers mod a . Then for any ε from the interval I^{m} = [1 − τ^{−m}, 1) we have

I^{m} ⊂ Att(S_{ε}(a, b)).  (4.24)

For each point x ∈ I and its orbit Orb_{ε}(x) consisting of the elements S_{ε}^{j}(a, b)(x), j = 0, 1, 2, …, the equality

(Orb_{ε}(x) ∩ I^{m})^{c} = I^{m}  (4.25)

is true. In equality (4.25) we denote by the index c a closure operation .

□

4.4. Lower derivatives of the composite shift. It is easy to see that the composite shift S_{ε}(a, b) (4.7) has the symmetry

S_{ε}(a, b)(x) = S_{inv(ε)}(−a, −b)(inv(x))  (4.26)

that corresponds to the central symmetry

(x_{1}, x_{2}) ⟼ (inv(x_{1}), inv(x_{2}))

of the unit square I^{c} ×I^{c}, where inv(x) = 1−x is an involution of the unit interval I^{c} = [0, 1]. The definition of upper derivatives S_{ε}(a, b)^{(m)} (see (4.8)) is based on the graduation of I by the differentiation intervals I^{m} = [ ε_{m}, 1) for ε_{m} = 1 − τ^{−m}. In the graduation let us replace the intervals I^{m} by their images

I_{m} = inv(I^{m}) = ( 0, 1 − ε_{m}] = ( 0, τ^{−m}]

under the involution inv. Then equality (4.8) transforms to the definition of lower derivatives of the composite shift

S_{ε}(a, b)_{(m)} = S_{ε}(a, b)|I_{m} × I_{m}.  (4.27)

It means a restriction of the parameters x and ε onto the interval I_{m}. Symmetry (4.26) connects lower derivatives (4.27) and upper derivatives (4.8) by the equality

S_{ε}(a, b)_{(m)}(x) = S_{inv(ε)}(−a, −b)^{(m)}(inv(x)).  (4.28)

Using Theorem 1 we can prove the following result for the lower derivative S_{ε}(a, b)_{(m)}.

Theorem 2. Suppose the composite shift S_{ε}(a, b) (see (4.7)) of the unit interval I has the integer parameters a, b of the same sign. Let the congruences

F_{m+1} ≡ F_{m+2} ≡ F mod b  (4.29)

be true, where F_{m+1}, F_{m+2} are the Fibonacci numbers, and 1 ≤ F ≤ |b|. Suppose |a| ≤ s(a′, m) for a′ = (|a|F^{−1})_{b} and the function s(a′, m) (see (4.19),(4.12)). Under these conditions the following statements hold for the lower derivatives S_{ε}(a, b)_{(m)} (see (4.27)).

For even m,

S_{ε}(a, b)_{(m)} ∼ S_{ε_{′}} (sign(a) · a′, b)  (4.30)

with the renormalization

ε_{′} = ε/(1 − ε_{m});  (4.31)

for odd m,

S_{ε}(a, b)_{(m)} ∼ S_{ε_{′}} (−sign(a) · a′, −b)  (4.32)

with the same renormalization ε_{′} of the parameter ε.

Proof. Let’s calculate the lower derivative S_{ε}(a, b)_{(m)} with the help of equality (4.28) and Theorem 1. In the case of an even level m, by formula (4.18), we obtain the similarity for the upper derivative in (4.28)

S_{inv(ε)}(−a, −b)^{(m)}(inv(x)) ∼ S_{inv(ε)′}(−b, sign(−a) · a′)(inv(x))  (4.33)

with the renormalization (see (4.19))

inv(ε)′ = \frac{inv(ε) − ε_{m}}{1 − ε_{m}} = 1 − \frac{ε}{1 − ε_{m}} .

Thus, we get the conjugate formula

inv(ε)′ = inv(ε_{′})  (4.34)

of upper renormalizations (4.19) and lower renormalizations (4.31) of the parameter ε ∈ I. Using this formula and symmetry (4.26) of the composite shift , we obtain the equality

S_{inv(ε)′}(−b, sign(−a) · a′)(inv(x)) =
= S_{inv(ε_{′})}(−b, sign(−a) · a′)(inv(x)) = S_{ε_{′}} (sign(a) · a′, b).

On the other hand, by formula (4.28), the upper derivative in (4.33) is equal to S_{ε}(a, b)_{(m)}(x). Therefore, for even m, similarity (4.30) follows from (4.33) and the last equality.

In the case of an odd level m the upper derivative in equality (4.28) can be calculated by formula (4.20). This completes the proof.

□

For the same reason, with the help of symmetries (4.26) and (4.28), the selfsimilarity property in Corollaries 1 - 3 is transferred to the lower derivatives of the composite shift S_{ε}(a, b)_{(m)} (see (4.27)).

5. Internal derivatives and plateaus

5.1. Lacunas. Consider the composite shift S_{ε}(a, b) (see (4.7)) with the integer parameters 1 ≤ b < a, (a, b) = 1. Suppose the followings congruences are fulfilled

F_{m+1} ≡ b mod a, F_{m+2} ≡ 0 mod a.  (5.1)

From the Fibonacci tiling T il(ε_{m}) of an even level m we allocate a subset Orb(I^{m}) = Orb(G_{0}^{m}) + Orb(E_{0}^{m}). This set is a sum of (5.2) and (5.3), where

G_{0}^{m} → G_{b}^{m} → G_{b+a}^{m} → ⋯ → G_{b+na}^{m} → ⋯ → G_{F_{m+1}−a}^{m}  (5.2)

is the orbit Orb(G_{0}^{m}) of intervals G_{j}^{m} with indexes j = 0 and j = b + na, 0 ≤ n <

\frac{F_{m+1} − b}{a}, and

E_{0}^{m} → E_{a}^{m} → ⋯ → E_{na}^{m} → ⋯ → E_{F_{m+2}−a}^{m}  (5.3)

is the orbit Orb(E_{0}^{m}) of intervals E_{i}^{m} with indexes i = na, 0 ≤ n < \frac{F_{m+2}}{a}. For the intervals from orbits (5.2) and (5.3) we assume that the inclusions

G_{0}^{m} ⊂ I_{ε}^{+}, G_{j}^{m} ⊂ I_{ε}^{−} for j > 0,  (5.4)
E_{i}^{m} ⊂ I_{ε}^{−} for all i

are true, where I_{ε}^{−} = [ 0, ε) and I_{ε}^{+} = [ ε, 1). Then from the definition of the composite shift S_{ε}(a, b) (see (4.7)) it follows that the set Orb(I^{m}) is the disjoint sum of the S_{ε}(a, b)-orbits of the intervals G_{0}^{m} and E_{0}^{m}. By equality (4.2) we get

S_{ε}(a, b)(Orb(I^{m})) = Orb(I^{m}),  (5.5)

i.e. the set Orb(I^{m}) is invariant under the transformation S_{ε}(a, b).

A lacuna L^{m} is the greatest possible length interval L^{m} ⊂ I that touches on the left the interval G_{0}^{m} and disjoints with the set Orb(I^{m}). Our aim is to find the lacuna L^{m} of the orbit Orb(I^{m}).

We apply an Algorithm from [10] for building the Fibonacci tilings T il(ε_{m}). This algorithm uses quasi-local coordinates for the generating intervals .

△ From the right to the left we build the sequence

… G_{k^{∗}}^{m}, E_{k^{∗}}^{m}, G_{k}^{m}, E_{k}^{m}, …, G_{0}^{m}, E_{0}^{m}  (5.6)

which consists of the intervals G_{l}^{m}, E_{l}^{m} with the indexes k = 0, 1, …, F_{m+2} − 1, by the following rule: If m is even, the transition k ↦ k^{⋆} in (5.6) is

k ↦ k^{⋆} = k + F_{m} mod F_{m+2}

and from sequence (5.6) all intervals G_{k}^{m} with indexes k ≥ F_{m+1} are deleted. If m is odd, the transition k ↦ k^{⋆} is

k ↦ k^{⋆} = k + F_{m+1} mod F_{m+2}

and from sequence (5.6) the intervals E_{k}^{m} with indexes k ≥ F_{m+1} are deleted. ▽

Using Algorithm (5.6), we can write the lacuna L^{m} for an even level m as the sum

L^{m} = E_{−(a−1)b}^{m} ⊕ G_{−(a−2)b}^{m} ⊕ E_{−(a−2)b}^{m} ⊕ ⋯ ⊕ G_{−2b}^{m} ⊕ E_{−2b}^{m} ⊕ G_{−b}^{m} ⊕ E_{−b}^{m} (5.7)

of the intervals that touch each other. These intervals E_{k}^{m} have indexes

k ≡ sF_{m} ≡ −bs mod F_{m+2},  (5.8)

where 1 ≤ s ≤ a − 1. Under condition (5.1) the parameters a, b are coprime and a divides F_{m+2}. Hence, k ≡ −bs ≢ 0 mod a. According to (5.6) the short intervals G_{k}^{m} from decomposition (5.7) have indexes

k ≡ tF_{m} ≡ −bt mod F_{m+2}, k < F_{m+1},  (5.9)

where 1 ≤ t ≤ a − 2. In this case k ≡ −bt ≢ b mod a, as for the same reasons the congruence −bt ≡ b mod a is equivalent to t ≡ a − 1 mod a.

From Algorithm (5.6) it follows that the interval L^{m} (see (5.7)) touches on the left G_{0}^{m}. We also see that L^{m} disjoints with the set Orb(I^{m}) under conditions (5.4). The interval L^{m} has the maximal possible length with preservation of the specified properties. The lacuna L^{m} contains a − 1 intervals E_{k}^{m} and

t(a, m) = ♯\{1 ≤ t ≤ a − 2 : \{\frac{tF_{m}}{F_{m+2}} \} < \frac{F_{m+1}}{F_{m+2}} \}  (5.10)

intervals G_{k}^{m}. It has the length

|L^{m}| = (a − 1)|E_{0}^{m}| + t(a, m)|G_{0}^{m}|.  (5.11)

In the case of an odd level m we consider the set Orb(I^{m}) = Orb(E_{0}^{m})+Orb(G_{0}^{m}) consisting of the sets

\begin{matrix}E_{0}^{m} → E_{b}^{m} → E_{b+a}^{m} → ⋯ → E_{b+na}^{m} → ⋯ → G_{F_{m+1}−a}^{m}, \\ G_{0}^{m} → G_{a}^{m} → ⋯ → G_{na}^{m} → ⋯ → G_{F_{m+2}−a}^{m} \end{matrix}  (5.12)

that are the orbits Orb(E_{0}^{m}) and Orb(G_{0}^{m}). The intervals E_{j}^{m} and G_{i}^{m} have indexes (5.2) and (5.3), respectively.

Assume that the intervals from orbits (5.12) satisfy the conditions

E_{0}^{m} ⊂ I_{ε}^{+}, E_{j}^{m} ⊂ I_{ε}^{−} for j > 0,

G_{i}^{m} ⊂ I_{ε}^{−} for all i. (5.13)

In these conditions the set Orb(I^{m}) is equal to the sum of the S_{ε}(a, b)-orbits of the intervals E_{0}^{m}, G_{0}^{m}. Orb(I^{m}) is an invariant set (see (5.5)) under the composite shift S_{ε}(a, b).

If m is odd, the lacuna L^{m} is situated on the right-side of the interval E_{0}^{m}. By Algorithm (5.6) we write L^{m} as the sum

L^{m} = G_{−b}^{m} ⊕ E_{−b}^{m} ⊕ G_{−2b}^{m} ⊕ E_{−2b}^{m} ⊕ ⋯ ⊕ G_{−(a−2)b}^{m} ⊕ E_{−(a−2)b}^{m} ⊕ G_{−(a−1)b}^{m} (5.14)

of the intervals G_{0}^{m} and E_{0}^{m} with indexes (5.8) and (5.9), respectively. Therefore, the length of L^{m} is equal to

|L^{m}| = (a − 1)|G_{0}^{m}| + t(a, m)|E_{0}^{m}|,  (5.15)

where t(a, m) is the function in (5.10).

Now let us find when inclusions (5.4) and (5.13) hold. We require an embedding of the Fibonacci tiling T il(ε_{m}) to the unit interval I = [0, 1). This embedding depends on the parameters m and ε and is defined by the equation

T il(ε_{m}, δ) = t_{δ}(T il(ε_{m})).  (5.16)

Here

I ⟶^{t_{δ}} I : x ⟼ x + δ mod 1

is the simple shift of the interval I. If m is even, we choose

δ = |E_{0}^{m}|, ε ∈ J^{m} = [1 − |G_{0}^{m}| − |L^{m}|, 1 − |G_{0}^{m}|).  (5.17)

In the case of odd m we assume the following restrictions on the parameters δ and ε

δ = ε − (1 − |E_{0}^{m}|), ε ∈ J^{m} = [1 − |E_{0}^{m}| − |L^{m}|, 1 − |E_{0}^{m}|).  (5.18)

In both cases, according to equalities (2.6), (2.7), (5.11), and (5.15), the length of the lacuna L^{m}(δ) is equal to

|L^{m}(δ)| = (a − 1)τ^{−(m+1)} + t(a, m)τ^{−(m+2)}.  (5.19)

Embending t_{δ} (5.16) generates the set

Orb(I^{m}(δ)) = Orb(G_{0}^{m}(δ)) + Orb(E_{0}^{m}(δ))

for the shifted interval I^{m}(δ) = t_{δ}(I^{m}) = G_{0}^{m}(δ) + E_{0}^{m}(δ). It is easy to see that for ε ∈ J^{m} the intervals from the orbits Orb(G_{0}^{m}(δ)) and Orb(E_{0}^{m}(δ)) satisfy conditions (5.4), (5.13). Hence, the specified set is an orbit of the interval I^{m}(δ) under the shift S_{ε}(a, b).

Now we need to introduce the concept of integrated transformation

T_{ε}^{±} = ∫ S^{±}d_{ε}

acting on the space Î_{ε} = {(x, i) : x ∈ I, 0 ≤ i < d_{ε}(x)} by the rule

T_{ε}^{±} = \begin{cases}(x, i + 1) & if i + 1 < d_{ε}, \\ (S^{±}(x), 0) & if i + 1 = d_{ε}. \end{cases}

Here S^{±1} is the simple shift x ↦ x±τ mod 1 of the unit interval I. The distribution function d_{ε} : I → ℕ = {1, 2, …} is equal to

d_{ε}(x) = \begin{cases}orb(G_{0}^{m}(δ)) & if x ∈ [0, ⟨−τ⟩), \\ orb(E_{0}^{m}(δ)) & if x ∈ [⟨−τ⟩, 1) \end{cases}

for even m, and

d_{ε}(x) = \begin{cases}orb(G_{0}^{m}(δ)) & if x ∈ [0, ⟨τ⟩), \\ orb(E_{0}^{m}(δ)) & if x ∈ [⟨τ⟩, 1) \end{cases}

for odd m.

According to (5.2), (5.3), (5.12) the number of various intervals in the orbits Orb(G_{0}^{m}(δ)) and Orb(E_{0}^{m}(δ)) is equal to

orb(G_{0}^{m}(δ)) = 1 + \frac{F_{m+1} − b}{a}, orb(E_{0}^{m}(δ)) = \frac{F_{m+2}}{a}

for an even level m, and

orb(G_{0}^{m}(δ)) = \frac{F_{m+2}}{a} , orb(E_{0}^{m}(δ)) = 1 + \frac{F_{m+1} − b}{a}

for odd m, respectively.

Proposition 3. Let S_{ε}(a, b) be composite shift (4.7) with the integer coprime parameters 1 ≤ b < a satisfying the congruences F_{m+1} ≡ b mod a, F_{m+2} ≡ 0 mod a. Then we have the following isomorphisms

S_{ε}(a, b)|I^{m}(δ) × J^{m} ≃ S^{±1},  (5.20)
S_{ε}(a, b)|Orb(I^{m}(δ)) × J^{m} ≃ ∫ S^{±1}d_{ε}  (5.21)

for the restrictions of the composite shift S_{ε}(a, b) for the parameter x onto the interval I^{m}(δ) = G_{0}^{m}(δ) + E_{0}^{m}(δ) and its orbit Orb(I^{m}(δ)), and for the parameter ε onto the interval J^{m} (5.17), (5.18). In the right-hand side (5.20) and (5.21) there is the simple shift S : x ↦ x + τ mod 1 or its inverse shift S^{−1} : x ↦ x − τ mod 1 for m even or odd, respectively.

Proof. As above, for ε from the interval J^{m} (5.17), (5.18) the S_{ε}(a, b)-orbits Orb(I^{m}(δ)) preserve the structure of orbits Orb(I^{m}) (5.2), (5.3) and (5.12). Thus, isomorphisms (5.20) and (5.21) follow from the similarity property (see (4.4) and (4.6)) for the derivatives S|I^{m} ∼ S^{±1}.

□

5.2. Spirals. By Proposition 3, the set Orb(I^{m}(δ)) is invariant under the action of the composite shift S_{ε}(a, b) with the parameter ε ∈ J^{m}. Now we need to prove that Orb(I^{m}(δ)) is an attraction area for the unit interval I and the dynamic distance from any point x ∈ I to the set Orb(I^{m}(δ)) is finite. It is enough to prove both properties for the points x from the spiral set

Spir_{ε} = I^{m}\Orb(I^{m}(δ)).  (5.22)

Explanation of this name for the supplementation Spir_{ε} of the set Orb(I^{m}(δ)) follows from the proof of the following result.

Proposition 4. Suppose composite shift S_{ε}(a, b) (4.7) satisfies the conditions in Proposition 3. We conjecture that the parameter ε belongs to the interval J^{m} (see (5.17), (5.18)). Then for any point x from the unit interval I there exists a number k = 0, 1, 2, … for which S_{ε}^{k}(a, b)(x) visits the set Orb(I^{m}(δ)) which is the S_{ε}(a, b)- orbit of the interval I^{m}(δ) from Subsection 5.1. The number k depends on the parameters ε, a, b.

Proof. Let us recall the conditions from proposition 3 for the parameters a and b:

\begin{matrix}1 ≤ b < a, & (a, b) = 1, \\ b ≤ F_{m+1}, & a ≤ F_{m+2}. \end{matrix}  (5.23)

As F_{m+1} ≡ b mod a, and F_{m+1} > 0, we obtain that F_{m+1} = b+ka for some integer k ≥ 0. Thus, the third condition in (5.23) holds. The fourth condition in (5.23) holds, because the Fibonacci number F_{m+2} is divided themselves on a.

Lacunas L^{m} (5.7) and (5.14) are symmetric for even and odd levels m. Therefore, without loss of generality we can consider only the case of even m.

Using decomposition (5.7) and embedding t_{δ} (5.16), we write the lacuna L^{m}(δ) as the sum

L^{m}(δ) = E_{k_{a−1}}^{m}(δ) ⊕ G_{k_{a−2}}^{m}(δ) ⊕ E_{k_{a−2}}^{m}(δ) ⊕ ⋯ ⊕ G_{k_{1}}^{m}(δ) ⊕ E_{k_{1}}^{m}(δ).  (5.24)

Here the intervals E_{k}^{m}(δ) have the indexes (see (5.8))

k_{s} = sF_{m} = −bs mod F_{m+2}, 1 ≤ s ≤ a − 1.

By (5.9), we impose the additional restrictions

k_{s} < F_{m+1}, 1 ≤ s ≤ a − 2,

for the indexes of the intervals E_{k}^{m}(δ).

Now we show that any ε from interval J^{m} (5.17) belongs to one of the intervals E_{k_{s}}^{m}(δ) or G_{k_{s}}^{m}(δ) from decomposition (5.24). We use the induction on s = 1, 2, …, a−1.Assume that the parameter ε runs the intervals E_{k_{1}}^{m}(δ), G_{k_{1}}^{m}(δ), …, E_{k_{a−1}}^{m}(δ) in the above order.

Let’s accept the following agreement. The shift S_{ε}(a, b) translates any point x in the interval I_{ε}^{−} = [0, ε) to the point x′ ≡ x + aτ mod 1, where a = 1, 2, . . .. Since τ = (1 + \sqrt{5})/2 is irrational, then the point x visits the interval I_{ε}^{+} = [ε, 1) = I\I_{ε}^{−} through a finite number of iterations. Without loss of generality, we can assume that the point x in Proposition 4 belongs to the set

Spir_{ε} ∩ I_{ε}^{+} = L^{m}(δ) ∩ I_{ε}^{+},

where Spir_{ε} is the spiral set (see (5.22)).

Case E. Suppose s = 1, ε ∈ E_{k_{1}}^{m}(δ), and the initial point x belongs to E_{k_{1}}^{m}(δ)∩I_{ε}^{+}. According to (5.8) the index k_{1} is equal to F_{m} . By (5.23), we obtain k_{1}+b ≤ F_{m+2}. The following two cases are possible. Case 1) k_{1}+b < F_{m+2}. Case 2) k_{1}+b = F_{m+2}.

In the first case the point x is translated by the shift S_{ε}(a, b) to a point x′ from the interval E_{k′_{1}}^{m}(δ) with the index k′_{1} = k_{1} + b. As k′_{1} ≡ 0 mod a, then using (5.3) we see that the point x′ visits the orbit Orb(I^{m}(δ)).

In the second case, from the tiling G_{0}^{m} ⊕ E_{0}^{m} = E_{F_{m+2}}^{m} ⊕ G_{F_{m+1}}^{m} (see (4.2)) it follows that x is translated to a point x′ in the interval I^{m}(δ) = G_{0}^{m} ⊕ E_{0}^{m} so that x′ again visits the orbit Orb(I^{m}(δ)).

Therefore, in both cases the composite shift S_{ε}(a, b) moves the point x through the intervals which form the chains

\begin{matrix}E_{k_{1}}^{m}(δ) ⟹ E_{k′_{1}}^{m}(δ) ⊂ Orb(I^{m}(δ)), \\ E_{k_{1}}^{m}(δ) ⟹ I^{m}(δ) ⊂ Orb(I^{m}(δ)). \end{matrix}  (5.25)

Here we denote by the double arrows the shift from the interval I_{ε}^{+}.

Case GE. Assume that s = 1, ε ∈ G_{k_{1}}^{m}(δ), and the initial point x belongs to the set G_{k_{1}}^{m}(δ) ∩ I_{ε}^{+}. If the point x belongs to the previous set E_{k_{1}}^{m}(δ) ∩ I_{ε}^{+} that is equal to E_{k_{1}}^{m}(δ), its orbit (5.25) does not change. Let us consider two cases 1) k_{1} + b < F_{m+1}, and 2) k_{1} + b ≥ F_{m+1}.

In the first case the point x is translated to a point x′ in the interval G_{k′_{1}}^{m}(δ) with the index k′_{1} = k_{1} + b < F_{m+1}, k′_{1} ≡ 0 mod a. By the condition in Proposition 4 F_{m+1} ≡ b mod a, hence there exists the least integer t_{min} ≥ 0 satisfying the inequalities b + at_{min} < F_{m+1} and k_{1} + b + at_{min} ≥ F_{m+1}. By (4.2), the point x′ moves to a point x′′ in the interval E_{k′′_{1}}^{m}(δ) with the index

k′′_{1} = k′_{1} + at_{min} − F_{m+1} = k_{1} + (b + at_{min} − F_{m+1}) ≤ k_{1}.

From k1^{′′} ≡ k_{1} mod a we conclude that the point x′′ is translated to a point x′′′ in the previous interval E_{k_{1}}^{m}(δ).

In the second case, by (4.2), we see that the point x′ belongs to the interval E_{k′_{1}}^{m}(δ) with the index k′_{1} = k_{1} + (bF_{m+1}) ≤ k_{1}. Hence, x′ is translated to a point x′′ again in the interval E_{k_{1}}^{m}(δ). In the case GE the composite shift S_{ε}(a, b) moves the point x through the intervals which form the chains

\begin{matrix}G_{k_{1}}^{m}(δ) ⟹ G_{k′_{1}}^{m}(δ) ⟶ E_{k′′_{1}}^{m}(δ) ⟶ E_{k_{1}}^{m}(δ), \\ G_{k_{1}}^{m}(δ) ⟹ E_{k′_{1}}^{m}(δ) ⟶ E_{k_{1}}^{m}(δ). \end{matrix}  (5.26)

Here we denote by the double arrows and the simple arrows shifts from the intervals I_{ε}^{+} and I_{ε}^{−}, respectively.

Let us remark that proof of the induction step s ↦ s + 1 for s = 1, 2, …, a − 2 depends on the fact

E_{k_{s+1}}^{m}(δ) ⊕ G_{k_{s}}^{m}(δ), G_{k_{s+1}}^{m}(δ) ⊕ E_{k_{s+1}}^{m}(δ) ⊕ G_{k_{s}}^{m}(δ)  (5.27)
E_{k_{s+1}}^{m}(δ) ⊕ E_{k_{s}}^{m}(δ), G_{k_{s+1}}^{m}(δ) ⊕ E_{k_{s+1}}^{m}(δ) ⊕ E_{k_{s}}^{m}(δ)  (5.28)

whether the lacuna L^{m}(δ) contains the interval G_{k_{s}}^{m}(δ) (see (5.27)) or not(see (5.28)).

In the case EG the parameter ε belongs to the interval E_{k_{s+1}}^{m}(δ). The initial point belongs to the intersection E_{k_{s+1}}^{m}(δ) ∩ I_{ε}^{+}. As the lacuna L^{m}(δ) contains the interval G_{k_{s}}^{m}(δ) (5.27), we have k_{s} < F_{m+1} and k_{s+1} = k_{s} + F_{m} < F_{m+2}. Let us consider two cases 1) k_{s+1} + b < F_{m+2}, and 2) k_{s+1} + b ≥ F_{m+2}.

In the first case the point x is translated to a point x′ in the interval E_{k′_{s+1}}^{m}(δ) with the index k′_{s+1} = k_{s+1} + b = k_{s} + F_{m} + b. Therefore, there exists t_{min} ≥ 0 so that b + at_{min} ≤ F_{m+1} k_{s} + F_{m} + b + at_{min} ≥ F_{m+2}. Hence, by (4.2), we obtain that the point x′ is translated by the shift S_{ε}(a, b) to a point x′′ which belongs to the sum G_{k′′_{s+1}}^{m}(δ) ⊕ E_{k′′_{s+1}}^{m}(δ). Here the intervals G_{k′′_{s+1}}^{m}(δ) and E_{k′′_{s+1}}^{m}(δ) touch each other and have the index k′′_{s+1} = k_{s} + F_{m} + b + at_{min} − F_{m+2}. By the choice of t_{min} , we have k′′_{s+1} = k_{s} + (b + at_{min} − F_{m+1}) ≤ k_{s}. Since ks+1^{′′} ≡ k_{s} mod a, the point x′′ is translated to x′′′ in the interval sum G_{k_{s}}^{m}(δ) ⊕ E_{k_{s}}^{m}(δ).

In the second case the point x at once gets in the set G_{k′_{s+1}}^{m}(δ) ⊕ E_{k′_{s+1}}^{m}(δ), where the index k′_{s+1} = k_{s+1} + b − F_{m+2} = k_{s} + (b − F_{m+1}) is less than or equal to k_{s}.

In any case the initial point x moves through the following chains

\begin{matrix}E_{k_{s+1}}^{m}(δ) ⟹ E_{k′_{s+1}}^{m}(δ) ⟶ G_{k′′_{s+1}}^{m}(δ) ⊕ E_{k′′_{s+1}}^{m}(δ) ⟶ G_{k_{s}}^{m}(δ) ⊕ E_{k_{s}}^{m}(δ), \\ E_{k_{s+1}}^{m}(δ) ⟹ G_{k′_{s+1}}^{m}(δ) ⊕ E_{k′_{s+1}}^{m}(δ) ⟶ G_{k_{s}}^{m}(δ) ⊕ E_{k_{s}}^{m}(δ). \end{matrix} (5.29)

By the scheme (5.27), in the case GEG the parameter ε lies in G_{k_{s+1}}^{m}(δ), x lies in the set G_{k_{s+1}}^{m}(δ)∩I_{ε}^{+}. Since ε lies in the interval J^{m} (see (5.17)), then s ≤ a−3. As in the above case the index k_{s+1} is equal to k_{s} + F_{m}. The existence of the interval G_{k_{s+1}}^{m}(δ) in the lacuna L^{m}(δ) implies the inequality k_{s+1} < F_{m+1}. Consider two cases: 1) k_{s+1} + b < F_{m+1}, 2) k_{s+1} + b ≥ F_{m+1}.

In the first case the point x is translated to a point x′ in the interval G_{k′_{s+1}}^{m}(δ) with the index k′_{s+1} = k_{s+1} + b. Further, we choose t_{min} ≥ 0 under the conditions b + at_{min} ≤ F_{m+1}, k_{s+1} + b + at_{min} ≥ F_{m+1}. Then the point x′ moves to a point x′′ in the interval E_{k′′_{s+1}}^{m}(δ) with the index ks+1^{′′} = k^{{′′ \atop s+1}} + at_{min} − F_{m+1}. This index

satisfies the inequality k^{{′′ \atop s+1}} = k_{s+1} +(b+at_{min} −F_{m+1}) ≤ k_{s+1} and the congruence

ks+1^{′′} ≡ k_{s+1} mod a.

In the second case the point x′ visits the interval E_{k′_{s+1}}^{m}(δ). By (5.24), the index

k^{{′′ \atop s+1}} = k_{s+1} + b − F_{m+1} satisfies the inequality k^{{′′ \atop s+1}} ≤ k_{s+1}.

The moving ways of the initial point x look as following:

\begin{matrix}G_{k_{s+1}}^{m}(δ) ⟹ G_{k′_{s+1}}^{m}(δ) ⟶ E_{k′′_{s+1}}^{m}(δ) ⟶ E_{k_{s+1}}^{m}(δ), \\ G_{k_{s+1}}^{m}(δ) ⟹ E_{k′_{s+1}}^{m}(δ) ⟶ E_{k_{s+1}}^{m}(δ). \end{matrix}  (5.30)

To complete the proof of Proposition 4 we need to consider the induction step in two cases (5.28).

In the case EE we have ε in the interval E_{k_{s+1}}^{m}(δ). An initial point x belongs to the set E_{k_{s+1}}^{m}(δ) ∩ I_{ε}^{+}. Since the lacuna L^{m}(δ) does not contain the interval G_{k_{s}}^{m}(δ), we obtain k_{s} ≥ F_{m+1}. Hence, the next index is equal to k_{s+1} = k_{s} + F_{m} − F_{m+2} = k_{s}F_{m+1}. The index k_{s+1} is less than or equal to F_{m+2}. Therefore, it follows k_{s+1} = k_{s} − F_{m+1} < F_{m}. By (5.24), we obtain that the initial point x moves to a point x′ in the interval E_{k′_{s+1}}^{m}(δ) with the index k′_{s+1} = k_{s+1} + b = k_{s} + (b − F_{m+1}). This index is less than or equal to k_{s}, and k′_{s+1} ≡ k_{s} mod a. Thus, we conclude that the moving of the initial point x is possible only along an unique chain

E_{k_{s+1}}^{m}(δ) ⟹ E_{k′_{s+1}}^{m}(δ) ⟶ E_{k_{s}}^{m}(δ).  (5.31)

In the last case GEE from (5.28) it follows ε ∈ G_{k_{s+1}}^{m}(δ) and the initial point x belongs to the intersection G_{k_{s+1}}^{m}(δ) ∩ I_{ε}^{+}. For the same reasons, as in the case EE, the index k_{s+1} is less than F_{m}. Hence, the interval G_{k_{s+1}}^{m}(δ) belongs to the lacuna L^{m}(δ). Consider two cases: 1) k_{s+1} + b < F_{m+1}, and 2) k_{s+1} + b ≥ F_{m+1}.

In the first case the point x is translated by the shift S_{ε}(a, b) to a point x′ in the interval G_{k′_{s+1}}^{m}(δ) with the index k′_{s+1} = k′_{s+1} + b. Further we choose t_{min} ≥ 0 under the conditions b + at_{min} ≤ F_{m+1} and k_{s+1} + b + at_{min} ≥ F_{m+1}. Then the next point x′′ lies in the interval E_{k′′_{s+1}}^{m}(δ) with the index

k′′_{s+1} = k′_{s+1} + at_{min} − F_{m+1} = k_{s+1} + (b + at_{min} − F_{m+1}) ≤ k_{s+1}.

This index is congruent to the index k_{s+1} mod a.

In the second case we have k^{{′′ \atop s+1}} = k_{s+1} + (b − F_{m+1}) ≤ k_{s+1}. Hence, the moving of the initial point x is possible only along two chains

\begin{matrix}G_{k_{s+1}}^{m}(δ) ⟹ G_{k′_{s+1}}^{m}(δ) ⟶ E_{k′′_{s+1}}^{m}(δ) ⟶ E_{k_{s+1}}^{m}(δ), \\ G_{k_{s+1}}^{m}(δ) ⟹ E_{k′_{s+1}}^{m}(δ) ⟶ E_{k_{s+1}}^{m}(δ). \end{matrix}  (5.32)

By (5.25) - (5.32), we obtain the following result. For any point x in the set L^{m}(δ) ∩ I^{+}, the beginning of the S_{ε}(a, b)-orbit x → x′ → x′′ → ⋯ does not depend on the choice of the parameter ε in the interval J^{m} (see (5.17)). Using this property of orbits, we complete the proof of Proposition 4.

□

5.3. Frequency distribution and plateaus. Suppose that ε lies in the interval J^{m}. By Proposition 4, we have the equality

Orb(I^{m}(δ)) = Att(S_{ε}(a, b)).  (5.33)

Here Att(S_{ε}(a, b)) is the attractor of the composite shift S_{ε}(a, b) from Section 4.3. By Proposition 4, the attractor has the property

Att(S_{ε}(a, b)) = ⋂_{0≤k≤k(ε,a,b)} S_{ε}^{k}(a, b)(I).

Applying these equalities and Proposition 3, we obtain the following theorem.

Theorem 3. Suppose the composite shift S_{ε}(a, b) (see (4.7)) satisfies the conditions in Proposition 3. Suppose the parameter ε lies in the interval J^{m} (see (5.17), (5.18)). Then for any x, θ in the unit interval I = [ 0, 1) there exists the limit

ν_{ε}(θ) = lim_{n→∞} \frac{1}{n} ♯\{k : S_{ε}^{k}(a, b)(x) ≤ θ, k = 0, 1, …, n − 1\}

that is equal to the frequency of which points in the S_{ε}(a, b)-orbit of x visit the interval [0, θ]. The limit ν_{ε}(θ) does not depend on the choice of the initial point x ∈ I and ν_{ε}(θ) can be calculated by the formula

ν_{ε}(θ) = \frac{|Att(S_{ε}(a, b)) ∩ [0, θ]|}{|Att(S_{ε}(a, b))|},  (5.34)

where | · | is the Haar measure on the torus I ≃ ℝ/ℤ.

□

Further we shall be interested in the diagonal frequency ν_{ε} = ν_{ε}(θ). We write (5.34) as the following formula

ν_{ε} = \frac{|Att(S_{ε}(a, b))^{−}|}{|Att(S_{ε}(a, b))|},  (5.35)

where Att(S_{ε}(a, b))^{−} is equal to the intersection of the attractor Att(S_{ε}(a, b)) with the interval I_{ε}^{−} = [0, ε) in Section 5.1.

Consider an infinite series of the composite shifts S_{ε}(F_{n}, F_{n−1}) for n ≥ 3. The sequence of the Fibonacci numbers F_{1}, F_{2}, …, F_{m}, … mod F_{n} has the period (see (3.13))

T (F_{n}) = \begin{cases}3, & if n = 3, \\ 2k, & if n ≥ 4, n even, \\ 4k, & if n ≥ 4, n odd. \end{cases}  (5.36)

Therefore, the congruences

F_{m+1} ≡ F_{n−1} mod F_{n} , F_{m+2} ≡ 0 mod F_{n}

hold if and only if the level m of the Fibonacci tiling T il(ε_{m}) belongs to the progression

m ≡ n − 2 mod T (F_{n}).  (5.37)

The orbits Orb(I^{m}(δ)) (see (5.2), (5.3), (5.12)), the lacunas L^{m}(δ) (see (5.7), (5.14)), and the internal derivatives of the composite shifts S_{ε}(F_{n}, F_{n−1}) (see (5.20), (5.21)) are symmetric for even and odd levels m. Therefore, without loss of generality, we can assume that m is even.

The length of the lacuna L^{m}(δ) is calculated by formula (5.10)

|L^{m}(δ)| = (F_{n} − 1)τ^{−(m+1)} + t(F_{n}, m)τ^{−(m+2)}.

Here, by (5.10), the function t(F_{n}, m) is equal to

t(F_{n}, m) = ♯\{1 ≤ t ≤ F_{n} − 2 : \{\frac{tF_{m}}{F_{m+2}} \} < \frac{F_{m+1}}{F_{m+2}} \}.

Let us prove that the value of the function t(F_{n}, m) does not depend on the parameter m, if m satisfy the condition in (5.37). If m = n − 2, we arrive at the formula

t(F_{n}, n − 2) = ♯\{1 ≤ t ≤ F_{n} − 2 : \{\frac{tF_{n−2}}{F_{n}} \} < \frac{F_{n−1}}{F_{n}} \}.

Let us calculate t(F_{n}, n − 2) with the help of the Algorithm in Section 5.1 that builds the Fibonacci tilings T il(ε_{m}). First, expand the interval 0 ≤ t ≤ F_{n} − 1. Then the right part in the equality for t(F_{n}, n − 2) becomes equal to F_{n−1}. It is easy to see that the inequality for the fractional parts holds, if t = 0, and it does not hold, if t = F_{n} − 1. Thus, we obtain the equality

t(F_{n}, n − 2) = F_{n−1} − 1

for n ≥ 3.

The Fibonacci tiling T il(ε_{m}) consists of F_{m+1} short intervals and F_{m+2} long intervals. In the paper [10] the author proved that the Fibonacci tilings T il(ε_{m}) satisfy the recurrence formula

T il^{+}(ε_{m+2}) = τ^{−1} · \overline{T il^{+}(ε_{m+1})} ⊕ τ^{−2} · T il^{+}(ε_{m})

of the sort as for the Fibonacci numbers F_{m}, i.e. F_{m+2} = F_{m+1} +F_{m}. The line over \overline{T il^{+}(ε_{m+1})} indicates the inversion of designations for long and short intervals. We denote by ⊕ the non-commutative operation that puts the first tiling to the second tiling compressed by τ^{−1} and τ^{−2} times, respectively.

By the construction of lacunas L^{m}, the Algorithm in Section 5.1, and the recurrence formula, we get that the function t(F_{n}, m) does not depend on the parameter m in (5.37). Using the specified invariancy and the equality for the function t(F_{n}, n − 2), we obtain the formula

t(F_{n}, m) = F_{n−1} − 1.  (5.38)

This formula takes place for all n ≥ 3 and m ≡ n − 2 mod T (F_{n}). With the help of formula (5.38), we find the lacuna length

|L^{m}(δ)| = τ^{−m}(τ^{n−2} − 1).  (5.39)

By the definition of the orbit Orb(I^{m}) (see (5.2), (5.3)), we can write its measure as

|Orb(I^{m}(δ))| = |G_{0}^{m}(δ)| · orb(G_{0}^{m}(δ)) + |E_{0}^{m}(δ)| · orb(E_{0}^{m}(δ)),

where the factors

orb(G_{0}^{m}(δ)) = 1 + \frac{F_{m+1} − F_{n−1}}{F_{n}} , orb(E_{0}^{m}(δ)) = \frac{F_{m+2}}{F_{n}}

are equal to the numbers of intervals in the orbits Orb(G_{0}^{m}(δ)) and Orb(E_{0}^{m}(δ)), respectively. The lengths |G_{0}^{m}(δ)| = |G_{0}^{m}|and |E_{0}^{m}(δ)| = |E_{0}^{m}| can be calculated by formulas (2.6). By equality (5.33), we find the attractor measure

|Att(S_{ε}(F_{n}, F_{n−1}))| = \frac{1}{F_{n}} (1 + F_{n−2}τ^{−(m+2)})  (5.40)

for the composite shift S_{ε}(F_{n}, F_{n−1}), if its parameter ε lies in the interval (see (5.17))

J^{m} = [1 − τ^{−(m+2)} − τ^{−m}(τ^{n−2} − 1), 1 − τ^{−(m+2)}).  (5.41)

Using the equal

Att(S_{ε}(F_{n}, F_{n−1}))^{−} = Att(S_{ε}(F_{n}, F_{n−1}))\{G_{0}^{m}(δ)}

and (5.40), we have

\begin{aligned}|Att(S_{ε}(F_{n}, F_{n−1}))^{−}| & = \frac{1}{F_{n}} (1 + F_{n−2}τ^{−(m+2)}) − τ^{−(m+2)} = \\ & = \frac{1}{F_{n}} (1 − F_{n−1}τ^{−(m+2)}). \end{aligned}  (5.42)

Proposition 5. Let S_{ε}(F_{n}, F_{n−1}) be composite shift (4.7). Suppose the parameter ε lies in the interval J^{m} (see (5.41)), where m ≡ n−2 mod T (F_{n}), n ≥ 3, and T (F_{n}) is the period determined by formula (5.36). Then for all initial points x in the unit interval I = [ 0, 1) the frequency ν_{ε} of which points in the S_{ε}(F_{n}, F_{n−1})-orbit of x visit the interval I_{ε}^{−} = [ 0, ε) is calculated by the formula

ν_{ε} = \frac{1 − F_{n−1}τ^{−(m+2)}}{1 + F_{n−2}τ^{−(m+2)}}  (5.43)

or by the formula

ν_{ε} = 1 − \frac{τ^{−(m+2)}}{|Att(S_{ε}(F_{n}, F_{n−1}))|}  (5.44)

which connects the frequency ν_{ε} with the attractor measure |Att(S_{ε}(F_{n}, F_{n−1}))| (see (5.40)).

Proof. Substituting equality (5.40) and the second equality from (5.42) in formula (5.35) for the diagonal frequency ν_{ε}, we prove formula (5.43). Using (5.40) and the first equality from (5.42), we get formula (5.44).

□

Using formulae (5.43) and (5.44), we get

Corollary 5. Under the conditions in Propositon 5, the attractor measure |Att(S_{ε}(F_{n}, F_{n−1}))| is equal to

|Att(S_{ε}(F_{n}, F_{n−1}))| = \frac{τ^{−(m+2)}}{1 − ν_{ε}} .

So (cf. (5.40))

|Att(S_{ε}(F_{n}, F_{n−1}))| → \frac{1}{F_{n}}

as ε → 1 and ε lies in the union of the intervals J^{m}.

Consider two special cases of formula (5.43): 1) m − n − 2 and 2) n = 3.

Corollary 6. Suppose the parameter ε of the composite shift S_{ε}(F_{n}, F_{n−1}), n ≥ 3, lies in the interval J^{n−2} = [τ^{−(n−1)}, 1 − τ^{−n}). Then the frequency ν_{ε} does not depend on the number n of the shift series and ν_{ε} is equal to

ν_{ε} = τ^{−1} = 0.618 …  (5.45)

Corollary 7. Suppose the parameter ε of the composite shift S_{ε}(2, 1) lies in the interval J^{m} = [1 − τ^{−m}, 1 − τ^{(m+2)}) of the level m ≡ 1 mod 3, m ≥ 1. Then the frequency ν_{ε} is calculated by the following formula (see Fig.2 )

ν_{ε} = th_{τ} (\frac{m}{2} + 1).  (5.46)

Here the function th_{τ} (x) = (τ^{x} − τ^{−x})/(τ^{x} + τ^{−x}) is the hyperbolic tangent with the basis equal to the golden section τ = (1 + \sqrt{5})/2.

6. Spectral decomposition and singularities

6.1. The composite shift S_{ε}(3, 1). We apply the differentiation method from Subsections 4.4 to researching the structure of the composite shift S_{ε}(3, 1). By Corollary 2, we have the self-similarity property

S_{ε}(3, 1)^{(8)} = S_{ε}(3, 1)|I^{8} × I^{8} ∼ S_{ε′}(3, 1),  (6.1)

where I^{8} = [ε_{8}, 1) and ε_{8} = 1 − τ^{−8}. Therefore, without loss of generality, we can consider a behaviour of the shift S_{ε}(3, 1) only for ε from the fundamental area F = [0, ε_{8}) of the unit interval I = [0, 1). Various types of behaviour of the shift S_{ε}(3, 1) correspond to the intervals F_{1}, …, F_{4} from the tiling

F = F_{1} ⊕ F_{2} ⊕ F_{3} ⊕ F_{4} =
= [0, τ^{−1}) ⊕ [τ^{−1}, ε_{4}) ⊕ [ε_{4}, ε_{4} + τ^{−7}) ⊕ [ε_{4} + τ^{−7}, ε_{8}).

Since the congruences F_{7} ≡ 1 mod 3, F_{8} ≡ 0 mod 3 are fulfilled, we apply Proposition 3 for even m = 6 to the shift S_{ε}(3, 1) . Accordingly (5.17) the interval J^{6} coincides with F_{4}. Hence, by formula (5.21), we obtain the isomorphism

S_{ε}(3, 1)|Orb(I^{6}(δ)) × F_{4} ≃ ∫ S d_{ε}.  (6.2)

In the integral there is the distribution function

d_{ε} = \begin{cases}5, & if x ∈ [0, ⟨−τ⟩), \\ 7, & if x ∈ [⟨−τ⟩, 1). \end{cases}

For any ε ∈ F_{4} and any initial point x ∈ I we denote by ν_{ε} the frequency that points from the S_{ε}(3, 1)-orbit of x visit the interval I_{ε}^{−} = [0, ε). By Theorem 3, the frequency ν_{ε} is calculated by the formula

ν_{ε} = \frac{|Att(S_{ε}(3, 1))^{−}|}{|Att(S_{ε}(3, 1))|},

Using equality (5.33) and formulas (2.6), we see that the above measures are equal to

\begin{matrix}| Att(S_{ε}(3, 1)) | = |Orb(I^{6}(δ))| = 5τ^{−8} + 7τ^{−7}, \\ |Att(S_{ε}(3, 1))^{−}| = 4τ^{−8} + 7τ^{−7}. \end{matrix}

Hence, for these ε and x the frequency ν_{ε} is constant. It is equal to

ν_{ε} = 0.9397 …  (6.3)

By Theorem 1 for the level m = 4 and Theorem 2 for the level m = 3, we calculate the mixed derivative

S_{ε}(3, 1)| F_{3} × F_{3} = (S_{ε}(3, 1)^{(4)})_{(4)} ∼ S_{ε”}(−1, −2)  (6.4)

of the composite shift S_{ε}(3, 1).

Let us show that the behaviour of the shift S_{ε}(3, 1), when the parameter ε belongs to the intervals F_{1} and F_{2}, differs from the above behaviour of the shift on the intervals F_{3} and F_{4}, .

If ε ∈ F_{2}, we use the embedded Fibonacci tiling T il(ε_{2}, δ) of the level m = 2 (see (5.16)), where the shift δ is equal to |E_{0}^{2}| = τ^{−3}. We find the internal derivative

S_{ε}(3, 1)|Att(S_{ε}(3, 1)) × F_{2} ∼ S(\frac{τ}{2} ).  (6.5)

Here

\begin{matrix}Att(S_{ε}(3, 1)) = T il(ε_{2}, δ)\\{E_{1}^{2}(δ)\} = I\F_{2}, \\ Spir_{ε} = \{E_{1}^{2}(δ)\} = F_{2}, \end{matrix}

and S (\frac{τ}{2} ) is the shift of the unit interval I → I : x ↦ x + \frac{τ}{2} mod 1. Since

\begin{matrix}|Att(S_{ε}(3, 1))| = 2τ^{−3} + 2τ^{−4}, \\ |Att(S_{ε}(3, 1))^{−}| = 2τ^{−3} + τ^{−4}, \end{matrix}

the frequency ν_{ε} does not depend on the parameter ε from the interval F_{2}, and ν_{ε} is equal to

ν_{ε} = 0.809 …  (6.6)

In the case ε ∈ F_{1}, we apply the tiling T il(ε_{1}, δ) of the level m = 1 with δ = −|GE_{0}^{1}| = −τ^{−2}. Then we obtain

\begin{matrix}S_{ε}(3, 1)|F_{1} × F_{1} ∼ S_{ε′}(−2, −1), \\ S_{ε}(3, 1)| I × F_{1} ≃ ∫ S_{ε′}(−2, −1) d_{ε′}. \end{matrix}  (6.7)

Here x ∈ I, ϵ ∈ F_{1}, ε′ = τ · ε is a renormalization of the parameter ε, and d_{ε′} : I × I → ℕ is the distribution function

d_{ε′}(x) = \begin{cases}1 & if (x, ε′) ∈ I × I\D, \\ 2 & if (x, ε′) ∈ D, \end{cases}

where D is a non-crossing sum of two rectangulars

[0, τ^{−1}) × [0, τ^{−3}), [0, τ^{−3}) × [τ^{−3}, 1)

and the triangle

{(x, ε′) : x ∈ [ε′, τ^{−1}), ε′ ∈ [τ^{−3}, τ^{−1})}.

6.2. Spectrum. For any composite shift S_{ε}(a, b) (see (4.7)) we define the function μ(ν) for ν ∈ [0, 1) by the equality

μ(ν) = |{ε ∈ [0, 1) : ν_{ε} = ν}|.

The spectrum sp(a, b) is the set of all ν ∈ [0, 1) for which the weights μ(ν) > 0.

It is possible to show that for the composite shift S_{ε}(2, 1) formula (5.46) includes all spectrum sp(2, 1). This spectrum consists of the points

ν^{m} = th_{τ} (\frac{m}{2} + 1)

with the numbers m ≡ 1 mod 3, m ≥ 1, and the weights μ^{m} = τ^{−(m+1)}. Thus, each spectral point ν^{m} ∈ sp(2, 1) corresponds to a plateau of the frequency function ν_{ε}. The weight μ^{m} = μ(ν^{m}) is equal to the length of an interval for which the function ν_{ε} is constant. Taking into account that

ν^{m} ∼ 1 − τ^{−(m+2)} as m → ∞,

we obtain the asymptotic self-similarity property for the graph of the function μ(ν) (see Fig. 3). It means that the function μ(ν) is asymptotic invariant under the homothetic transformation with center at the point (ν, μ) = (1, 0) and ratio τ^{−3} . We write this self-similarity property of the shift S_{ε}(2, 1) by the symbol

sp(2, 1) = ( |).  (6.8)

Similarly to (6.8), from (6.3), (6.4), (6.6), and (6.7) it follows that the composite shift S_{ε}(3, 1) has the spectral decomposition

sp(3, 1) = (| (2, 1)|(−1, −2) |).

Here the first vertical line corresponds to a limiting point of the spectrum sp(3, 1), the second line corresponds to its discrete point (see Fig. 5).

Here is the list of spectral decompositions for composite shifts S_{ε}(a, b) with the integer coprime parameters a, b, satisfying the conditions 1 ≤ b < a ≤ 4:

\begin{matrix}sp(3, 1) = (| (2, 1)|(−1, −2) |), \\ sp(3, 2) = (−1, −2) | (3, 1), \\ sp(3, 2) = ((−1, −2) | (2, 1)|), \\ sp(4, 1) = ((2, 1)|(−1, −2) |), \\ sp(4, 3) = ((2, 3)|(−1, −2) | (2, 1)|(3, 2)|). \end{matrix}  (6.9)

The above list does not contain the composite shift S_{ε}(4, 2) with not coprime parameters a = 4, b = 2. The shift S_{ε}(4, 2) can be interpreted as a composite shift with the coprime parameters a = 2, b = 1. For this aim we need to replace τ with 2τ in definition (4.7). The shift S_{ε}(4, 2) belongs to the α-family of composite shifts

S_{ε}(a, b)(x) = x + aα mod 1 if x ∈ I_{ε}^{−},
S_{ε}(a, b)(x) = x + bα mod 1 if x ∈ I_{ε}^{+}, (6.10)

where α = 2τ. To research these composite shifts we require a tiling T il of the unit interval I = [0, 1) that corresponds to the irrationality α = 2τ (cf. Section 2 ).

6.3. Singularities. Let sp(a, b)^{c} be a closure of the spectrum sp(a, b) by a topology of the tours I ∼ ℝ/ℤ. For example, the set sp(2, 1)^{c} (see (6.8)) consists of the discrete points ν^{m} and one limiting point s ≡ 0 mod 1. The last point is a singularity of the degree deg(s) = 1. Discrete points of a spectrum we consider as singularities of zero degree.

The closure sp(3, 1)^{c} (see (6.9)) is more difficultly arranged. It includes the discrete points of the spectrum sp(3, 1) (ν = 0.9387 … (6.3), and others ), the singularities of the first degree (ν = 0.809 … (6.6) etc.), and the singularity s ≡ 0 mod 1 of the second degree. By self-similarity property (6.1), the singularity of the second degree is a limiting point of the singularities of the degree 1.

Let’s define the degree of a singularity by the induction on deg. Then we see that the singularity s ≡ 0 mod 1 in the spectrum closure sp(4, 1)^{c} (see (6.9)) has the degree 3.

The degree deg(a, b) of the spectrum sp(a, b) is equal to the maximal degree of singularities of the composite shift S_{ε}(a, b). For the composite shift S_{ε}(a, b), the degree deg(a, b) characterizes complexity of the spectral decomposition (see (6.9) or the length of monotone chains

S_{ε}(a_{1}, b_{1}) ↪ S_{ε}(a_{2}, b_{2}) ↪ ⋯ ↪ S_{ε}(a_{g}, b_{g}) ↪ S_{ε}(a, b).

Proposition 6. Let S_{ε}(a, b) be a composite shift (see (4.7))with the integer coprime parameters 1 ≤ b < a ≤ F_{n+2}. Then for any n = 1, 2, 3, … there exists the shift S_{ε}(a, b) with the spectral degree

deg(a, b) ≥ n.

Remark 1. In Proposition 6 the lower estimate deg(a, b) ≥ n is not exact, because the composite shift S_{ε}(5, 1) has the degree deg(5, 1) = 4.

Proof. Let us prove that the composite shift S_{ε}(a, b) with the parameters a = F_{n+2}, b = F_{n+1} has the spectral degree deg(a, b) ≥ n. By Theorem 2 we calculate the lower derivative

S_{ε}(F_{n+2}, F_{n+1})_{(2T (F_{n+1}))} ∼ S_{ε′}(F_{n}, F_{n+1})  (6.11)

with the order equal to the double period T (F_{n+1}) (see (5.36)) of the sequence of the Fibonacci numbers mod F_{n+1}. Then, applying Theorem 1, we get the upper derivative

S_{ε′}(F_{n}, F_{n+1})^{(2T (F_{n}))} ∼ S_{ε′′}(F_{n}, F_{n−1}).

In the same way we come to one of the shifts S_{ε}(F_{2}, F_{3}) or S_{ε}(F_{3}, F_{2}), if n is even or odd, respectively. By Corollary 1, the initial shift S_{ε}(F_{n+2}, F_{n+1}) has the self-similarity property

S_{ε}(F_{n+2}, F_{n+1})^{(2T (F_{n+2}))} ∼ S_{ε′}(F_{n+2}, F_{n+1}).

Using this property and (6.11), we come to the following conclusion: if the derivative shift S_{ε′}(F_{n}, F_{n+1}) has a degree d_{n+1}, the initial shift S_{ε}(F_{n+2}, F_{n+1}) has the degree

d_{n+2} ≥ d_{n+1} + 1.

The composite shift S_{ε}(F_{2}, F_{3}) = S_{ε}(1, 2) and its inverse shift S_{ε}(F_{3}, F_{2}) = S_{ε}(2, 1) have the degree 1. Therefore, by the induction on n, we complete the proof of the proposition.

□

References

[1] Z. Coelho, A. Lopes, L.F. da Rocha, Absolutely continuous invariant measures for a class of affine interval exchange maps, Proc. Amer. Math. Soc. 123:11 (1995), 3533–3542.

[2] U. Grimm, M. Schreiber, Aperiogic tilings on the computer, arXiv: cond-math/9903010 (1999), 1-18.

[3] E. Hecke, Eber analytische Funktionen und die Verteilung von Zahlen mod. eins, Math. Sem. Hamburg. Univ. Bd. 1 (1921), 54-76.

[4] H. Kesten, On a cojecture of Erdös and Szüsz related to uniform distribution mod 1, Acta Arith. 14 (1973), 26-38.

[5] P. Liardet, Regularities of distribution, Compositio Math 61 (1987), 267-293.

[6] C. Oguey, M. Duneau, A. Katz, A geometrical approach of quasiperiodic tilings, Commun. Math. Phys. 118 (1988), 99-118.

[7] D. Wall, Fibonacci series modulo m, Amer. Math. Monthly 67 (1960), 525–532.

[8] V.G. Zhuravlev, Self-similar growth of periodic partitions and graphs, Algebra i Analis 13:2 (2001), 69-92 (in Russian), English transl. in St. Petersburg Math. J. 13:2 (2002), 201-220.

[9] V.G. Zhuravlev, Growth of random tilings and graphs: between crystal and chaos , Algebra i Analis 14:6 (2002), 129-168 (in Russian), English transl. in St. Petersburg Math. J. 14:6 (2003), 1-31.

[10] V.G. Zhuravlev, One-dimensional Fibonacci tilings, Izvestiya RAN: Ser. Mat.,1-57 , awaiting publication in 2004.

Vladimir State Pedagogical University, Verhny Dubrova 10-67, 600022 Vladimir, Russia

Current address: Max Planck Institute of Mathematics, Vivatsgasse 7, D-53111, Bonn, Germany

  E-mail address: zhuravl@mpim-bonn.mpg.de, vzhuravlev@mail.ru