DERIVATIVES OF CIRCLE ROTATIONS AND SIMILARITY OF ORBITS

VLADIMIR ZHURAVLEV, ANTON SHUTOV

Abstract. It is proved that a derivative of a circle rotation on any interval is either a circle rotation or a non-cyclic 3-interval exchange transformation. In the first case we calculate all possible values of new rotation angle. Also we prove that the restriction of an orbit of the circle rotation on proper intervals is similar to the orbit of other circle rotation.

1. Introduction

In the work we study arithmetics and geometry of an irrational rotation R_{α} = x → x − α (mod 1). A derivative of the transformation on a set Y is the first returning map, defined by the equality

d_{Y} R_{α} = R_{α}^{n_{Y} (x)}(x),

where

n_{Y} (x) = min{n ∈ ℕ : R_{α}^{N}(x) ∈ Y }.

The sets Y are intervals of the unit circle. The basic result of this article is the following theorem.

Theorem.Let R_{α} : x → x − α (mod 1) be a rotation on the unit circle. If I = I_{m} is a proper differentiation interval (see section 2), then the derivative d_{I}R_{α} is a circle rotation again. If I ≠ I_{m} is an improper interval, then d_{I}R_{α} is a noncyclic 3-interval exchange transformation.

Geometrical corollary of the above theorem is the similarity of orbits of circle rotations with different α. Denote by Orb(1, α, I_{0}) the sequence which consists of the elements −nα (mod 1), n = 0, 1, 2, . . .. A derivative d^{m}Orb(1, α, I_{0}) is the sequence with elements from Orb(1, α, I_{0}) ∩ I_{m} numbered in the order of their hitting in the proper interval I_{m}. Then the Theorem implies the existence of the homothety h_{m} with center at the point 1 such that

d^{m}Orb(1, α, I_{0}) = h_{m}Orb(1, d^{m}α, I_{0})

for some d^{m}α. Let α = [(q_{1}, ⋯, q_{r})] be a quadratic irrationality with purely periodic continued fraction expansion, and

T = \{ \begin{matrix}q_{1} + … + q_{r}, if r is even, \\ 2(q_{1} + … + q_{r}), if r is odd. \end{matrix}

Then

d^{T} α = α

and

d^{T} Orb(1, α, I_{0}) = h_{T} Orb(1, α, I_{0}).

So, in this case we have the self-similarity of the orbits.

1Date: 25 June, 2004.

Differentiation of orbits induces a map

α → d^{1}α

from the unit circle onto itself. This map has some interesting properties admitting natural interpretation in terms of symbolic dynamics. In section 5 we correspond to any α ∈ [0; 1) its code, i.e. the infinite sequence of zeroes and units. The action of the map d^{1} on this coding is equivalent to the action of the Bernoulli shift (see Theorem 5.2). It is unexpected that the constructed coding is the well-known decomposition of α into the Stern-Brocot sequence.

Historical remarks. The concept of differenciation map arose in the theory of dynamic systems as a discrete analogue of the Poincare sections. Rauzy [7] and Ferenczi [1],[2] began to study of circle rotation derivatives in connection with the bounded remainder sets problem. In this article we use the approach of Zhuravlev [9] based on the Fibonacci tilings. The article [9] contains the general theory of the Fibonacci tilings for τ = \frac{1+\sqrt{5}}{2}. Manuylov [6] transferred these results on the case τ_{2} = 1 + \sqrt{2}.

Acknowledgment. The author is grateful to the Max Planck Institute of Mathematics (Bonn) for hospitality during the preparation of this paper.

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

2. The generalized Fibonacci tilings

We define here generalized Fibonacci tilings and state some their properties. Proofs can be found in [9],[5],[8].

Let 𝕋 be a set of all tilings of the unit circle 𝕊^{1} = [0; 1) into a finite number of intervals. Let us define an operator

B : 𝕋 → 𝕋.

Its action is defined as follows. Let T il ∈ 𝕋 be a tiling of 𝕊^{1} which consists of a finite number of intervals. Let |X| be the least length of the tiling’s intervals. Then in the tiling B(T il) intervals from T il with the lengths |X| are kept. The intervals with the lengths |Y | > |X| splits into two intervals with the lengths |X| and |Y | − |X| respectively. In other words, the action of the B-operator means a putting off intervals of the least length on the left-hand side of all intervals in the tiling T il.

Suppose a tiling T il_{0}(α) consists of two intervals

T il_{0}(α) = [0; α) ⊕ [α; 1),

where we denote by ⊕ a noncommutative operation of applying of the intervals [0; α) and [α; 1).

The generalized Fibonacci tiling of the order m = 0, 1, 2, … is defined by the relation

T il_{m}(α) = B^{m}(T il_{0}(α)).

It is obvious T il_{m}(α) ≠ T il_{n}(α) for an irrational α , if m ≠ n. It is easy to show that the tiling T il_{m}(α) consists of intervals which have two different lengths. It also cannot ends by two intervals of the same length. Consider the map Col_{m,α}, corresponding to each interval of T il_{m}(α) color labels E and G under the following conditions.

1) Intervals T il_{m}(α) with the identical lengths have the same labels. Intervals of different lengths have different labels.

2) The extreme right interval in T il_{m}(α) has the label E.

As a result, we get the color Fibonacci tilings

CT il_{m}(α) = Col_{m,α}(T il_{m}(α))

for m = 0, 1, 2, … We will use the following notations:

L^{m}, S^{m} for long and short intervals in the tiling T il_{m}(α);

l_{m}(α), s_{m}(α) for lengths of intervals L^{m}, S^{m};

L_{m}(α), S_{m}(α) are cardinalities of the sets of intervals L^{m}, S^{m} in T il_{m}(α); E^{m}, G^{m} for E- and G-intervals in the tiling CT il_{m}(α); e_{m}(α), g_{m}(α) for for lengths of intervals E^{m}, G^{m} ; E_{m}(α), G_{m}(α) are cardinalities of the sets of intervals E^{m}, G^{m} in T il_{m}(α).

Everywhere, where it will not call ambiguities, the index α will be omitted.

To each interval in the tiling CT il_{m}(α) we correspond its number by the following rule. The tiling CT il_{0}(α) consist of two intervals

CT il_{0}(α) = G_{0}^{0} ⊕ E_{0}^{0} = [0; α) ⊕ [α; 1)

which have number 0. Further interval numbers are defined by the induction on m. If s_{m}(α) = e_{m}(α), then

\begin{matrix}E_{i}^{m} → E_{i}^{m+1}, \\ G_{i}^{m} → E_{i+E_{m}}^{m+1} ⊕ G_{i}^{m+1}. \end{matrix}

If s_{m}(α) = g_{m}(α), then

\begin{matrix}E_{i}^{m} → G_{i}^{m+1} ⊕ E_{i}^{m+1}, \\ G_{i}^{m} → G_{i+E_{m}}^{m+1}. \end{matrix}

We use the following facts about CT il_{m}(α).

Proposition 2.1. There is the equality

CT il_{m}(1 − α) = \overline{CT il_{m}(α)}^{(+)}.

Here we denote by ∗^{(+)} an operation of exchanging of two extreme right intervals in a tiling, \overline{CT il_{m}(α)} is a tiling obtaining from CT il_{m}(α) by replacement all E and G labels on G and E the labels, respectively.

Proposition 2.2.

\begin{matrix}⟨E_{m}α⟩ = g_{m}, \\ ⟨G_{m}α⟩ = 1 − e_{m}. \end{matrix}  (2.1)

Proposition 2.3.

\begin{matrix}E_{i}^{m} − α ≡ E_{i+1}^{m} (mod 1), \\ G_{i}^{m} − α ≡ G_{i+1}^{m} (mod 1). \end{matrix}  (2.2)

Proposition 2.4.

CT il_{m}(α) − α ≡ CT il_{m}(α)^{(+)} (mod 1).  (2.3)

3. derivatives of the circle rotations

Let X be a set and Y ⊂ X its subset. Let A : X → X be a map of the set X to itself. We define a derivative (or the first returning map)

d_{Y} A : Y → Y

by the relation

d_{Y} A = A^{n_{Y} (x)}(x),  (3.1)

where

n_{Y} (X) = min{n ∈ ℕ : A^{n}(x) ∈ Y }.

Let’s denote by R_{α} the rotation

R_{α} : x → x − α (mod 1)  (3.2)

of the unit cicle 𝕊^{1}. We are interested in the case when X = 𝕊^{1} is the unit circle, A = R_{α}, and Y = I is an interval in 𝕊^{1}. Without loss of generality, we can suppose that on the right-hand site of the interval I is a point 1.

We put I_{m} = G_{0}^{m} ⊕ E_{0}^{m}, i_{m} = |I_{m}| and call the intervals I_{m} as the proper differentiation intervals. First, we calculate the derivatives d_{I}R_{α} on the proper intervals I_{m}.

For this purpose we we consider the tiling CT il_{m}(α). From (2.2) it follows that under the action of the rotation R_{α} intervals in CT il_{m}(α) exchange by the following rules

\begin{matrix}G_{0}^{m} → G_{1}^{m} → … → G_{G_{m}}^{m}, \\ E_{0}^{m} → E_{1}^{m} → … → E_{E_{m}}^{m}. \end{matrix}  (3.3)

From Proposition 2.4 it follows that

I_{m} = G_{0}^{m} ⊕ E_{0}^{m} = E_{E_{m}}^{m} ⊕ G_{G_{m}}^{m}.

Thus, the action of d_{I_{m}}R_{α} is equivalent to exchanging of the intervals G_{0}^{m} and E_{0}^{m}. If we glue the right- and left-hand sides of the interval I_{m}, we obtain that d_{I_{m}}R_{α} is the map

I_{m} → I_{m} : x → x − g_{m} (mod I_{m}).  (3.4)

So, any derivative or restriction of a circle rotation onto the proper differenciation interval is a rotation of the circle again.

Now we consider an improper interval I. Assume that

I_{m+1} ⊂ I ⊂ I_{m}.  (3.5)

From (2.1) it follows that E_{0}^{m} ⊂ I. Let’s split the interval G_{0}^{m} into two intervals

G_{0}^{m} = G_{0,1}^{m} ⊕ G_{0,2}^{m}

such that I = G_{0,2}^{m} ⊕ E_{0}^{m}. From (3.5) it follows that |G_{0,1}^{m}| < |E_{0}^{m}|. We represent the interval E_{0}^{m} as the sum

E_{0}^{m} = E_{0,1}^{m} ⊕ E_{0,2}^{m},

where |E_{0,1}^{m}| = |G_{0,1}^{m}|. The map d_{I_{m}}R_{α} exchanges intervals by the following rules

G_{0,1}^{m} → G_{G_{m},1}^{m},
G_{0,2}^{m} → G_{G_{m},2}^{m},
E_{0,1}^{m} → E_{E_{m},1}^{m} = G_{0,1}^{m} → G_{G_{m},1}^{m},
E_{0,2}^{m} → E_{E_{m},2}^{m}.

Thus, we have

E_{E_{m},2}^{m} ⊕ G_{G_{m},1}^{m} ⊕ G_{G_{m},2}^{m} = E_{E_{m},2}^{m} ⊕ G_{G_{m}}^{m} = I,

as

|I| = e_{m} + g_{m} − |G_{0,1}^{m}|

and

|E_{E_{m},1}^{m}| + |G_{G_{m}}^{m}| = e_{m} + g_{m} − |G_{0,1}^{m}|.

We see that the derivative d_{I}R_{α} is the non-cyclic 3-intervals exchange transformation

G_{0,2}^{m} ⊕ E_{0,1}^{m} ⊕ E_{0,2}^{m} → E_{E_{m},2}^{m} ⊕ G_{G_{m},1}^{m} ⊕ G_{G_{m},2}^{m}

or, otherwise,

G_{0,2}^{m} ⊕ E_{0,1}^{m} ⊕ E_{0,2}^{m} → d_{I}R_{α}(E_{0,2}^{m}) ⊕ d_{I}R_{α}(E_{0,1}^{m}) ⊕ d_{I}R_{α}(G_{0,2}^{m}).

So, we obtain the following theorem.

Theorem 3.1. Let

R_{α} : x → x − α (mod 1)

be a rotation of the unit circle 𝕊^{1} = [0; 1). If I = I_{m} is a proper differentiation interval, then the derivative d_{I}R_{α} is a circle rotation again. If I ≠ I_{m} is an improper interval, then d_{I}R_{α} is a non-cyclic 3-interval exchange transformation.

The first part of this theorem can be reformulated using the orbit language. Let Orb(a, α, I) be an orbit of the point a under the map x → x − α (mod I). By definition, we have the following sequence of intervals

I_{0} ⊃ I_{1} ⊃ … ⊃ I_{m} ⊃ …

Let’s put S_{l} = (S, I_{l}), where S = (a_{i})_{i=0}^{∞} is a sequence in the interval I_{l},

d^{k}S_{l} = (S′, I_{l+k}),

where S′ = S ∩ I_{l+k} and elements of the sequence S′ = (a_{i_{j}})_{j=0}^{∞} are ordered by the induction:

i_{0} = min{i : a_{i} ∈ I_{l+k}},
i_{n+1} = min{i > i_{n} : a_{i} ∈ I_{l+k}}.

Theorem 3.2. Let

O_{m}^{−} = Orb(1, −g_{m}(α), I_{m}),
O_{m}^{+} = Orb(1, g_{m}(α), I_{m}),
Õ_{m}^{−} = Orb(1, −e_{m}(α), I_{m}),
Õ_{m}^{+} = Orb(1, e_{m}(α), I_{m}).

Then

d^{k}O_{m}^{−} = O_{m+k}^{−},
d^{k}O_{m}^{+} = O_{m+k}^{−},
d^{k}Õ_{m}^{−} = Õ_{m+k}^{−},
d^{k}Õ_{m}^{+} = Õ_{m+k}^{+}.

Proof. The first relation is actually proved in Theorem 3.1. The third relation follows from the first relation for 1 − α with the help of the equalities g_{m}(α) = e_{m}(1−α), e_{m}(α) = g_{m}(1−α). Other two relations follow from the proved relations with the help of the equality e_{m}(α) + g_{m}(α) = |I_{m}|.

Let h_{m} be a homothety with the center at the point 1, mapping the interval I_{m} to I_{0} = [0; 1). Consider the orbits O_{m} = h_{m}(O_{m}^{−}). Then

\begin{matrix}O_{0} = Orb(1, −α, 𝕊^{1}), \\ O_{m} = Orb(1, −d^{m}α, 𝕊^{1}) \end{matrix}

for the some d^{m}α. The map d^{m} has the property

d^{m_{1}}(d^{m_{2}}α) = d^{m_{1}+m_{2}}α.

From Theorem 3.2 it follows that

d^{m}α = \frac{g_{m}(α)}{g_{m}(α) + e_{m}(α)} = \frac{g_{m}(α)}{i_{m}(α)}.  (3.6)

Using (2.1), we may rewrite formula (3.6) as

d^{m}α = \frac{⟨E_{m}α⟩}{⟨E_{m}α⟩ − ⟨G_{m}α⟩ + 1}.

To receive explicit formulaes for d^{m}α is our further purpose.

4. Formulaes for d^{m}α

We obtain a formula for d^{1}α by direct calculation.

Proposition 4.1.

d^{1}α = \{ \begin{matrix}\frac{α}{1−α}, if α < \frac{1}{2}, \\ \frac{2α−1}{α}, if α > \frac{1}{2}. \end{matrix}

Proposition 4.2. Suppose that α = [0; q_{1}, q_{2}, q_{3}, …], where there is a continued fraction expansion of α in the square brackets. Then

d^{1}α = \{ \begin{matrix}[0; q_{1} − 1, q_{2}, q_{3}, …], if q_{1} > 1, \\ [0; 1, q_{2} − 1, q_{3}, …], if q_{1} = 1, q_{2} > 1, \\ [0; q_{3} + 1, q_{4}, q_{5}, …], if q_{1} = q_{2} = 1. \end{matrix}

To find d^{m}α we need to calculate e_{m}(α) and g_{m}(α). First we obtain formulaes for s_{m}(α) and l_{m}(α).

We note that

d^{m}α + d^{m}(1 − α) = 1.

Therefore we may consider only the case α ∈ (0; \frac{1}{2}).

Number i is called α-regular, if l_{m−1}(α) > l_{m}(α). Let {m_{i}} be a sequence of α-regular numbers,

n_{i} = m_{i} − m_{i−1},
\overline{s}_{i} = s_{m_{i}}(α),
\overline{l}_{i} = l_{m_{i}}(α).

The transition

T il_{m−1}(α) → T il_{m}(α)

transforms long and short intervals by the following rules:

\begin{matrix}S^{m} → S^{m+1}, \\ L^{m} → S^{m+1} ⊕ L^{m+1}, \end{matrix}

if m is not α-regular; and

\begin{matrix}S^{m} → L^{m+1}, \\ L^{m} → L^{m+1} ⊕ S^{m+1}, \end{matrix}

if m is α-regular. The transition

T il_{m_{i−1}}(α) → T il_{m_{i}}(α)

is defined by the substituons

\begin{matrix}S^{m_{i−1}} → L^{m_{i}}, \\ L^{m_{i−1}} → n_{i}L^{m_{i}} ⊕ S^{m_{i}}. \end{matrix}  (4.1)

From (4.1) it follows that

\overline{s}_{i} = \overline{l}_{i+1},  (4.2)
\overline{l}_{i−1} = n_{i}\overline{l}_{i} − \overline{l}_{i+1}.  (4.3)

Proposition 4.3. Let’s put G(α) = ⟨\frac{1}{α}⟩. Then

n_{1} = [\frac{1}{α}] − 1,

\overline{l}_{0} = 1 − α, \overline{s}_{0} = α.

For k ≥ 1 we have

n_{k} = [\frac{1}{G^{k−1}(α)}] ,  (4.4)

\overline{l}_{k}(α) = ∏_{i=0}^{k−1} G^{i}(α).  (4.5)

Proof. One can proof the Proposition by the induction on k using relations (4.2), (4.3), and the equalities

n_{k} = [\begin{matrix}\overline{l} & k \\ \overline{s} & k \end{matrix}] = [\begin{matrix}\overline{l} & k−1 \\ \overline{l} & k \end{matrix}].

The map G(α) = ⟨\frac{1}{α}⟩ is known as the Gauss-Kuzmin transformation [4]. The following properties of this map are well-known. Suppose that the continued fraction expansion of α is equal to α = [0; q_{1}, q_{2}, …, q_{n}, r_{n}(α)], where r_{n}(α) is a remainder. Then

q_{i} = [\frac{1}{G^{i−1}(α)}] ,  (4.6)
r_{n}(α) = G^{n}(α).  (4.7)

Here is supposed that G^{0}(α) = α.

Comparing (4.4) with (4.6) and (4.5) with (4.7), we see that

n_{k} = q_{k} for k > 1 and n_{1} = q_{1} − 1,  (4.8)
\overline{l}_{k} = ∏_{i=0}^{k−2} r_{i}(α).  (4.9)

We consider the problem of evaluation l_{i}(α) and s_{i}(α) in the case when m is not α-regular. Assume that m_{i−1} < m < m_{i}. We represent m as m = m_{i−1} + t, where 0 < t < n_{i}. A transition

T il_{m_{i−1}}(α) → T il_{m}(α)

8 VLADIMIR ZHURAVLEV, ANTON SHUTOV

is defined by the substitutions

\begin{matrix}S^{m_{i−1}} → S^{m}, \\ L^{m_{i−1}} → tS^{m} ⊕ L^{m}. \end{matrix}  (4.10)

Hence,

s_{m}(α) = \overline{s}_{i},  (4.11)
l_{m}(α) = \overline{l}_{i} − t\overline{s}_{i}.  (4.12)

Combining (4.2), (4.9) - (4.12), we get Proposition 4.4. Suppose that m = m_{i} + t, 0 ≤ t < n_{i+1}. Then

l_{m}(α) = ∏_{j=0}^{i} r_{j}(α),

s_{m}(α) = ∏_{j=0}^{i−1} r_{j}(α)(1 − tr_{i}(α)).

Secondly, we calculate e_{m}(α) and g_{m}(α). Now we need a formula for Col_{m,α}. As α ∈ (0; \frac{1}{2}), we have T il_{0}(α) = S^{0} ⊕ L^{0}. Hence,

\begin{matrix}Col_{0,α}(L) = E, \\ Col_{0,α}(S) = G. \end{matrix}  (4.13)

Formula (4.10) means that the transition

T il_{m_{i−1}}(α) → T il_{m_{i−1}+t}(α)

does not vary the last interval in the tiling. So, we get

Col_{m_{i−1}+t,α} ≡ Col_{m_{i−1},α}.  (4.14)

On the other hand, substitutions (4.1) imply

Col_{m_{i},α} ≡ \overline{Col_{m_{i−1},α}}.  (4.15)

Combining (4.13) - (4.15), we obtain Proposition 4.5. Suppose that m = m_{i} + t, 0 ≤ t < n_{i+1}. Then

Col_{m,α}(L) = \{ \begin{matrix}E, if i is even, \\ G, if i is odd, \end{matrix}

Col_{m,α}(S) = \{ \begin{matrix}G, if i is even, \\ E, if i is odd. \end{matrix}

The inverse map is equal to

Col_{m,α}^{−1}(E) = \{ \begin{matrix}L, if i is even, \\ S, if i is odd, \end{matrix}

Col_{m,α}^{−1}(G) = \{ \begin{matrix}S, if i is even, \\ L, if i is odd. \end{matrix}

(4.16)

Combining Propositions 4.4, 4.5 with formula (3.6), we obtain a formula for d^{m}α in terms of the continued fraction expansion of α. Proposition 4.6. Suppose that α ∈ (0; \frac{1}{2}), m = m_{i} + t, 0 ≤ t < n_{i+1}. Then

d^{m}α = \{ \begin{matrix}\frac{r_{i}(α)}{1+(1−t)r_{i}(α)}, if i is even, \\ \frac{1−tr_{i}(α)}{1+(1−t)r_{i}(α)}, if i is odd. \end{matrix}

5. Derivatives and Bernoulli shifts

We correspond to any number α the sequence

w(α) = (w_{0}, w_{1}, w_{2}, …),

where

w_{m} = \{ \begin{matrix}0, if g_{m}(α) < e_{m}(α), \\ 1, if g_{m}(α) > e_{m}(α). \end{matrix}

The sequence w(α) is finite only in the case g_{m}(α) = e_{m}(α). It is possible only for rational number α. From (4.16) it follows

w_{m} = \{ \begin{matrix}0, if i is even, \\ 1, if i is odd. \end{matrix}

for m = m_{i} + t, 0 ≤ t < n_{i+1}

We denote by

x^{n} = (x, …, x)

the sequence of n elements x. We see that a number α with the expansion α = [0; q_{1}, q_{2}, q_{3}, …] has the sequence

w(α) = 0^{q_{1}−1}1^{q_{2}}0^{q_{3}}1^{q_{4}} …  (5.1)

Therefore, the correspondence

α ↦ w(α)

between α ∈ [0; 1] and w(α) is one-to-one.

Proposition 5.1. {0, w(α)} is an expansion α in the Stern-Brocot sequence.

Proof. Let’s recall the definition of the Stern-Brocot sequence. For this purpose we consider two fractions \frac{0}{1} and \frac{1}{0}, and repeat the following operation:

insert the fraction \frac{m+m′}{n+n′} between two neigbour fractions \frac{m}{n} and \frac{m′}{n′}.

As a result, we get a tree of fractions which is called a Stern - Brocot tree. Binary searching of α in this tree gives an expansion of α in the Stern - Brocot sequence. Label 0 corresponds to left branches in the tree, but label 1 corresponds to right branches [3]. Comparing (5.1) with the expansion of α in the Stern - Brocot sequence, reduced in [3], we complete the proof of the Proposition.

Now we find the action of the map d^{1} on the sequence w(α).

Theorem 5.2. Suppose that w(α) = (w_{0}, w_{1}, w_{2}, …). Then

w(d^{1}α) = (w_{1}, w_{2}, w_{3}, …).

Thus, the action of the map d^{1} is equivalent to the Bernoulli shift on the set of 0/1-sequences.

Proof. By definition, we have

g_{0}(d^{1}α) = \frac{g_{1}(α)}{i_{1}(α)},

e_{0}(d^{1}α) = \frac{e_{1}(α)}{i_{1}(α)}.

Using the induction on m, it is easy to obtain that

g_{m}(d^{1}α) = \frac{g_{m+1}(α)}{i_{1}(α)},

e_{m}(d^{1}α) = \frac{e_{m+1}(α)}{i_{1}(α)}.

Hence, g_{m}(d^{1}α) > e_{m}(d^{1}α) if and only if g_{m+1}(α) > e_{m+1}(α). Similarly, g_{m}(d^{1}α) < e_{m}(d^{1}α) if and only if g_{m+1}(α) < e_{m+1}(α).

Corollary 5.3. For any α ∈ [0; 1) the equation

d^{m}β = α

has 2^{m} solutions.

Proof. For the proof it is enough to notice that w(β) = (w_{0}, …, w_{m−1}, w(α)), where (w_{0}, …, w_{m−1}) is one of 2^{m} possible 0/1 sequences.

Corollary 5.4. Suppose that α is irrational. Then the relation

d^{m+T} α = d^{m}α  (5.2)

holds for the some m and α if and only if α is a quadratic irrationality.

This corollary is equivalent to the fact that the Stern - Brocot sequence of α is periodic if and only if α is a quadratic irrationality. This fact is equivalent to the Lagrange theorem about continued fraction expansions of quadratic irrationalities.

The geometrical sence of relation (5.2) means a similarity of orbits and their derivatives. In other words, there exists a homothety h_{m,T} such that

h_{m,T} O_{m}^{−} = d^{T} O_{m}^{−}.  (5.3)

Using Theorem 3.2, we may rewrite relation (5.3) as the following relation

h_{m,T} O_{m}^{−} = O_{m+T}^{−} .

It means the similarity of the orbits.

If relation (5.2) is satisfied for some m = m_{0}, then it is also satisfied for all m > m_{0}. In this case the sequence of derivatives is periodic. Let T_{0}(α) be the least period.

Corollary 5.5. Let α be a quadratic irrationality. Suppose that the continued fraction exprasion of α is equal to

α = [0; q_{1}, …, q_{k}, (q_{k+1}, …, q_{k+r})].

Here in the interior brackets the period of the expansion is written. Then

T_{0}(α) = \{ \begin{matrix}q_{k+1} + … + q_{k+r}, if r is even, \\ 2(q_{k+1} + … + q_{k+r}), if r is odd. \end{matrix}

Proof. The proof is obvious.

Corollary 5.6. The number of all solutions of the equation T_{0}(α) = n is equal to

∑_{d|n} μ(\frac{n}{d})(2^{d} − 1).

Here μ(x) is the Möbius function.

Proof. The number of solutions of the equation d^{n}α = α is equal to 2^{n} − 1. It follows from the fact that the number of 0/1-sequences of the length n is equal to 2^{n}, and the sequence (1, …, 1) corresponds to α = 1. Further, it is enough to use the standard formula of transition from periodic points to primitive periodic points.

References

[1] S. Ferenczi, Bounded remainder sets, Acta Arithmetica, LXI.4,1992, pp.319-326

[2] S. Ferenczi, Ch. Holton, L. Zamboni, Structure of 3-interval exchange I: an arithmetic study, Preprints of Marcelle University, no 24,2001

[3] R. Graham, D. Knut, O. Patashnik Concrete mathematics, Addison-Wesley Pablishing, 1994.

[4] A.Ja. Khinchin, Continued fractions, Univ. Chicago Press, Chicago-London, 1964.

[5] N.N. Manuylov, The recurrent self-similar tilings, Chebyshev collection, 4, no.2,2003

[6] N.N. Manuylov, Self-similarity of some sequences, Zap. nauchn. seminars S-POMI, no.302, pp. 81-95

[7] G. Rauzy, Ensembles a restes bornes. Seminare de theorie des nombers de Bordeaux, 1983-1984, expose 24

[8] A.V. Shutov, N.N. Manuylov The global order of the circle tilings, Youth. Education. Economy. The collection of scientific papers of participants of 5-th All-Russia scientific - practical conference of young scientists, post-graduate students and students, Yaroslavl, 2004

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

Vladimir State Pedagogical University, Stroitelei 11, phis-mat, algebra

  E-mail address: vzhuravlev@mail.ru, shutov@vgpu.vladimir.ru

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

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