Dynamics of 2-interval piecewise affine maps and Hecke-Mahler series

TL;DR

This paper explicitly characterizes the dynamics of a specific class of 2-interval piecewise affine maps using functions related to Hecke-Mahler series, revealing rational rotation numbers for algebraic parameters.

Contribution

It provides an explicit description of the dynamics of 2-interval piecewise affine maps via functions involving Hecke-Mahler series, linking algebraic parameters to rational rotation numbers.

Findings

Explicit formulas for the dynamics using functions δ and φ

Rotation number is rational for algebraic parameters

Connections between dynamics and Hecke-Mahler series

Abstract

Let $f : [0,1)\rightarrow [0,1)$ be a $2$-interval piecewise affine increasing map which is injective but not surjective. Such a map $f$ has a rotation number and can be parametrized by three real numbers. We make fully explicit the dynamics of $f$ thanks to two specific functions $\delta$ and $\phi$ depending on these parameters whose definitions involve Hecke-Mahler series. As an application, we show that the rotation number of $f$ is rational, when the three parameters are algebraic numbers.

Equations

$$0<\lambda <1,\mu >0,1-\lambda <\delta <d_{\lambda ,\mu }:=\{\begin{array}{l}1if\lambda \mu <1,\\ \frac{\mu -\lambda \mu }{\mu -1},if\lambda \mu \ge 1.\end{array}$$

$$f(x)=\{\begin{array}{l}\lambda x+\delta ,if0\le x<\eta ,\\ \mu (\lambda x+\delta -1),if\eta \le x<1.\end{array}$$

$$\delta >\mu (\lambda +\delta -1)$$

$$\delta <\frac{\mu -\lambda \mu }{\mu -1}=d_{\lambda ,\mu },$$

$$f(x^{-})=\underset{y\nearrow x}{\lim }f(y)\mathrm{and}f(x^{+})=\underset{y\searrow x}{\lim }f(y),$$

$$\sigma =\sigma (\lambda ,\mu ,\rho ):=\sum _{k\ge 1}\left(\lfloor (k+1)\rho \rfloor -\lfloor k\rho \rfloor \right){\lambda }^k{\mu }^{\lfloor k\rho \rfloor }$$

$$\delta (\lambda ,\mu ,\rho )=\frac{(1-\lambda )(1+\mu \sigma )}{1+(\mu -1)\sigma }.$$

$$r_{\lambda ,\mu }=\{\begin{array}{l}1if\lambda \mu <1,\\ \frac{-\log \lambda }{\log \mu },if\lambda \mu \ge 1.\end{array}$$

$$\delta \left(\lambda ,\mu ,(p/q{)}^{-}\right)\le \delta \le \delta \left(\lambda ,\mu ,p/q\right)$$

$$\delta \left(\lambda ,\mu ,p/q\right)$$

$$\delta \left(\lambda ,\mu ,(p/q{)}^{-}\right)$$

$$S=S(\lambda ,\mu ,p/q):=\sum _{k=1}^{q-2}\left(\lfloor (k+1)p/q\rfloor -\lfloor kp/q\rfloor \right){\lambda }^k{\mu }^{\lfloor kp/q\rfloor }$$

$$\delta =\delta (\lambda ,\mu ,\rho ).$$

$${\varphi }_{\lambda ,\mu ,\delta ,\rho }(y)=\lfloor y\rfloor +\frac{1-\delta }{\lambda }+\sum _{k\ge 0}{\lambda }^k{\mu }^{\lfloor y\rfloor -\lfloor y-k\rho \rfloor }\left(\frac{\lambda +\delta -1}{\lambda }+\lfloor y-(k+1)\rho \rfloor -\lfloor y-k\rho \rfloor \right).$$

$$I\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\omega (x):=\bigcap _{n\to +\infty }\overline{\bigcup _{k\ge n}{f}^k(x)}$$

$$\delta \left(\lambda ,\mu ,(p/q{)}^{-}\right)\le \delta <\delta \left(\lambda ,\mu ,p/q\right).$$

$$C=\varphi (I)=\left\{\varphi (m/q);0\le m\le q-1\right\}$$

$${\Psi }_{\rho }(\lambda ,\mu )=$$

$${\Phi }_{\rho }(\lambda ,\mu ,x)=$$

$$\sigma (\lambda ,\mu ,\rho )=\sum _{k\ge 1}\left(\lfloor (k+1)\rho \rfloor -\lfloor k\rho \rfloor \right){\lambda }^k{\mu }^{\lfloor k\rho \rfloor }=\frac{1-\lambda }{\lambda \mu }{\Psi }_{\rho }(\lambda ,\mu ).$$

$${\Psi }_{\rho }(\lambda ,\mu )=\sum _{h\ge 1}\sum _{k\ge \frac{h}{\rho }}{\lambda }^k{\mu }^h=\sum _{h\ge 1}\sum _{k\ge \lceil \frac{h}{\rho }\rceil }{\lambda }^k{\mu }^h=\frac{1}{1-\lambda }\sum _{h\ge 1}{\lambda }^{\lceil \frac{h}{\rho }\rceil }{\mu }^h.$$

$$\sum _{h\ge 1}{\lambda }^{\lceil \frac{h}{\rho }\rceil }{\mu }^h=$$

$$=$$

$$\delta (\lambda ,\mu ,\rho )=\frac{(1-\lambda )(1+\mu \sigma (\lambda ,\mu ,\rho ))}{1+(\mu -1)\sigma (\lambda ,\mu ,\rho )}=\mu (1-\lambda )\frac{\lambda +(1-\lambda ){\Psi }_{\rho }(\lambda ,\mu )}{\lambda \mu +(1-\lambda )(\mu -1){\Psi }_{\rho }(\lambda ,\mu )}.$$

$$\underset{\rho \searrow 0}{\lim }{\Psi }_{\rho }(\lambda ,\mu )=0\mathrm{and}\mathrm{that}\underset{\rho \nearrow 1}{\lim }{\Psi }_{\rho }(\lambda ,\mu )=\frac{{\lambda }^2\mu }{(1-\lambda )(1-\lambda \mu )}.$$

$$0<{\Psi }_{\rho }(\lambda ,\mu )<\frac{{\lambda }^2\mu }{(1-\lambda )(1-\lambda \mu )}$$

$$x\mapsto \mu (1-\lambda )\frac{\lambda +(1-\lambda )x}{\lambda \mu +(1-\lambda )(\mu -1)x}$$

$$0<{\Psi }_{\rho }(\lambda ,\mu )<+\infty \mathrm{and}1-\lambda <\delta (\lambda ,\mu ,\rho )<\frac{\mu (1-\lambda )}{\mu -1}=d_{\lambda ,\mu }.$$

$$\sum _{k\ge 0}{\lambda }^k{\mu }^{\lceil k\rho +x\rceil }$$

$$\sum _{k\ge 0}(\lceil (k+1)\rho +x\rceil -\lceil k\rho +x\rceil ){\lambda }^k{\mu }^{\lceil k\rho +x\rceil }$$

$${\varphi }_{\lambda ,\mu ,\delta ,\rho }(y)=\lfloor y\rfloor +\frac{\delta }{1-\lambda }-\frac{\delta -\mu (\lambda +\delta -1)}{\lambda }{\Phi }_{\rho }(\lambda ,\mu ,-\{y\})$$

Full text

Michel Laurent and Arnaldo Nogueira

⁢⁢⁢⁢⁢⁢2010 Mathematics Subject Classification: 11J91, 37E05.

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1. Introduction

Definition 1**.**

Let $I=[0,1)$ be the unit interval. Let $\lambda,\mu,\delta$ be three real numbers. Assume

[TABLE]

Set $\eta={1-\delta\over\lambda}$ and define a map $f=f_{\lambda,\mu,\delta}:I\to I$ by the splited formula

[TABLE]

The restrictions of $f$ to the intervals $[0,\eta)$ and $[\eta,1)$ are increasing affine functions with slopes $\lambda$ and $\lambda\mu$, respectively. The symmetry $(x,y)\mapsto(1-x,1-y)$ on $[0,1]^{2}$ exchanges the two slopes. Our assumption that the first segment of the graph has a slope $\lambda$ less than $1$ is thus unrestrictive. Observe that $1-\lambda<d_{\lambda,\mu}\leq 1$ and that the bound $\delta\leq d_{\lambda,\mu}$ yields the injectivity of $f$. Indeed the inequality

[TABLE]

holds true for any $\delta$ in the interval $1-\lambda<\delta<1$ when $\lambda\mu<1$ ($f$ is then a piecewise contracting map) and is equivalent to

[TABLE]

when $\lambda\mu\geq 1$ (see Figure 1). Notice that in the limit case $\delta=d_{\lambda,\mu}$ with $\lambda\mu>1$, $f=f_{\lambda,\mu,\delta}$ becomes a bijection which was studied in [3].

We are concerned with the dynamics of the family of interval maps $f=f_{\lambda,\mu,\delta}$. We plan to relate their dynamics to the so-called Hecke-Mahler series in two variables $\lambda$ and $\mu$ (see Section 2 for definitions). The paper [16] deals with the case $\mu=1$ where the slope is constant. Although the map $f=f_{\lambda,\mu,\delta}$ is not necessarily a piecewise contraction, we extend part of the results established in [16] to a $2$-slope setting.

The dynamics of interval piecewise affine contractions has been studied by many authors, amongst others [6, 7, 8, 9, 11, 12, 13, 19, 22, 23]. According to [24], every map $f=f_{\lambda,\mu,\delta}$ has a rotation number $\rho=\rho_{\lambda,\mu,\delta}$, $0<\rho<1$. Although $f$ is not necessarily a piecewise contraction, we will prove that if $\rho$ takes an irrational value, then the closure $\overline{C}$ of the limit set $C:=\cap_{k\geq 1}f^{k}(I)$ of $f$ is a Cantor set and $f$ is topologically conjugated to the rotation map $x\in I\mapsto x+\rho\mod 1$ on C. When the rotation number is rational, the map $f$ has at most one periodic orbit (exactly one in most cases) and the limit set $C$ equals the periodic orbit when it does exist. More precisely, either $f$ or $f^{-}$ (a slight modification of the map $f$ whose definition is postponed to Section 8) has a periodic cycle.

We make the above mentioned qualitative results fully explicit thanks to formulae involving Hecke-Mahler series. Our approach is based on the study of a conjugation function $\phi$ which may be written down in terms of Hecke-Mahler series. The method was already performed in the special case $\mu=1$ where the two slopes are equal. It is motivated by Coutinho’s thesis [9] and has been recently reworked in [4, 14, 16]. The general case involving two different slopes is quite similar. Theorem 1 gives the value of the rotation number $\rho_{\lambda,\mu,\delta}$ in terms of the values of the three parameters $\lambda,\mu,\delta$, while Theorem 3 describes the behaviour of the orbits of $f$ and their relations with the conjugation $\phi$.

Next we introduce some standard notations. For any real function $f(x)$ of the real variable $x$, we denote by

[TABLE]

respectively, the left limit and the right limit of $f$ at $x$, whenever these limits do exist. As usual $\lfloor x\rfloor$ and $\lceil x\rceil$ stand, respectively, for the integer floor and the integer ceiling of the real number $x$. In particular, we have $\lceil x\rceil=\lfloor x\rfloor+1$ for any real number $x\notin{\mathbb{Z}}$ and $\lceil x\rceil=\lfloor x\rfloor$ when $x\in{\mathbb{Z}}$. We denote by $\{x\}=x-\lfloor x\rfloor$ the fractional part of $x$. The length of an interval $J\subset{\mathbb{R}}$ is denoted by $|J|$.

We first define a real function $\delta(\lambda,\mu,\rho)$ as follows.

Definition 2**.**

For positive real numbers $\lambda,\mu$ and $\rho$ such that $\lambda\mu^{\rho}<1$, set

[TABLE]

and

[TABLE]

For real numbers $\lambda$ and $\mu$ with $0<\lambda<1$ and $\mu>0$, set

[TABLE]

The series $\sigma(\lambda,\mu,\rho)$ converges when $0\leq\rho<r_{\lambda,\mu}$. For fixed $\lambda$ and $\mu$ with $0<\lambda<1,\mu>0$, the map $\rho\mapsto\delta(\lambda,\mu,\rho)$ is increasing in the interval $0\leq\rho<r_{\lambda,\mu}$ and it has a left discontinuity at each rational value (see Figure 2). It is continuous for any irrational $\rho$ and right continuous everywhere. The function $\delta$ enables us to compute the rotation number of $f_{\lambda,\mu,\delta}$ thanks to the

Theorem 1**.**

*Let $\lambda$ and $\mu$ be real numbers with $0<\lambda<1$ and $\mu>0$. Then the application $\delta\mapsto\rho_{\lambda,\mu,\delta}$ is a continuous non decreasing function sending the interval $(1-\lambda,d_{\lambda,\mu})$ onto the interval $(0,r_{\lambda,\mu})$ and satisfying the following properties:

(i) Let $p/q$ be a rational number with $0<p/q<r_{\lambda,\mu}$ where $p$ and $q$ are relatively prime integers. Then $\rho_{\lambda,\mu,\delta}$ takes the value $p/q$ if, and only if, $\delta$ is located in the interval*

[TABLE]

with the explicit formulae

[TABLE]

where

[TABLE]

*and the sum $S$ equals [math] when $q=2$.

(ii) For every irrational number $\rho$ with $0<\rho<r_{\lambda,\mu}$, there exists one and only one real number $\delta$ such that $1-\lambda<\delta<d_{\lambda,\mu}$ and $\rho_{\lambda,\mu,\delta}=\rho$ which is given by*

[TABLE]

Roughly speaking, the two maps $\rho\mapsto\delta(\lambda,\mu,\rho)$ and $\delta\mapsto\rho_{\lambda,\mu,\delta}$ are “inverse” from each other, meaning that their graphs are symmetric with respect to the main diagonal. In the special case $\mu=1$, we recover the formulae obtained in [16] for the map $f_{\lambda,1,\delta}$, which coincides with the contracted rotation $x\mapsto\{\lambda x+\delta\}$. Notice that the formulae of our Theorem 1 are consistent with those of Theorem 4.15 in [6], dealing with the subfamily of contractions $f_{\lambda,\mu,\delta}$ with $\lambda\mu<1$, although the formulations greatly differ.

Applying now a classical transcendence result, which is stated as Theorem 4 below, to the number $\delta(\lambda,\mu,\rho)$, we deduce from the assertion (ii) of Theorem 1 the following:

Theorem 2**.**

Let $\lambda,\mu,\delta$ be algebraic real numbers with $0<\lambda<1$ $\mu>0$ and $1-\lambda<\delta<d_{\lambda,\mu}$. Then, the rotation number $\rho_{\lambda,\mu,\delta}$ takes a rational value.

Notice that Theorem 2 no longer holds for the value $\delta=d_{\lambda,\mu}$ when $\lambda\mu>1$. Indeed, $d_{\lambda,\mu}={\mu-\lambda\mu\over\mu-1}$ is an algebraic number when $\lambda$ and $\mu$ are algebraic, while $f$ has rotation number $-\log\lambda/\log\mu$ by [3]. This ratio is a transcendental number when $\lambda$ and $\mu$ are non-zero algebraic numbers, unless $\lambda$ and $\mu$ are multiplicatively dependent.

We now investigate the behaviour of the iterates of $f=f_{\lambda,\mu,\delta}$ thanks to an explicit conjugation map $\phi$.

Definition 3**.**

Let $\lambda,\mu,\rho$ be three positive real numbers such that $\lambda\mu^{\rho}<1$, and let $\delta$ be an arbitrary real number. Let $\phi_{\lambda,\mu,\delta,\rho}:{\mathbb{R}}\to{\mathbb{R}}$ be the real function defined by the convergent series

[TABLE]

Theorem 3**.**

*Let $\lambda,\mu,\delta$ be three real numbers with $0<\lambda<1$, $\mu>0$ and $1-\lambda<\delta<d_{\lambda,\mu}$. Set $\rho=\rho_{\lambda,\mu,\delta}$ and $\phi=\phi_{\lambda,\mu,\delta,\rho}$.

(i) Assume that $\rho$ is irrational. Then $C=\phi(I)$ and the restriction of $f=f_{\lambda,\mu,\delta}$ to the invariant set $C$ is conjugate by $\phi$ to the rotation $R_{\rho}:y\mapsto y+\rho\,\hbox{ mod }\,1$. In other words, we have the commutative diagramm:*

[TABLE]

Moreover $\overline{\phi(I)}$ is a Cantor set and for every $x\in I$, the $\omega$-limit set

[TABLE]

*equals $\overline{C}$.

(ii) Assume that $\rho={p/q}$ is rational, where $p$ and $q$ are relatively prime, and that*

[TABLE]

Then

[TABLE]

is a cycle of order $q$ and we have the commutative diagramm:

where $R_{p/q}$ denotes the rotation $y\mapsto y+{p\over q}\,\hbox{ mod }\,1$. Moreover, for every $x\in I$, the $\omega$-limit set $\omega(x)$ equals $C$.

(iii) When $\delta=\delta\left(\lambda,\mu,{p\over q}\right)$, the limit set $C$ is empty and $\phi(I)$ is a finite set with $q$ elements containing $1$. For every $x\in I$, the $\omega$-limit set $\omega(x)$ coincides with $\phi(I)$.

The paper is organized as follows. In Section 2, we introduce Hecke-Mahler series and relate them to our functions $\delta$ and $\phi$. Then, Theorem 2 easily follows from Theorem 1. The purpose of Sections 3 and 4 is to establish the basic conjugation equations (1) and (3). This goal is achieved thanks to Lemma 4.2 where some relations connecting the parameter $\delta$ with values of the function $\delta(\lambda,\mu,\rho)$ are needed, as for instance the inequalities (2) in the case (3). It turns out that these constraints characterize the rotation number $\rho_{\lambda,\mu,\delta}$. As a consequence of the method, we establish Theorem 1 in Section 5. The next two sections provide additional information on the dynamics of $f$ in the case of an irrational rotation number (Proposition 5 in Section 6), or a rational one (Proposition 6 of Section 7). In both cases, we explicitly describe the iterated images $f^{n}(I),\,n\geq 1$. Finally Section 8 deals with the exceptional values of the form $\delta=\delta(\lambda,\mu,p/q)$ for which no periodic cycle exists.

2. Hecke-Mahler series and transcendental numbers

2.1. On Hecke-Mahler series.

We introduce the following sums:

Definition 4**.**

Let $\lambda$, $\mu$ and $\rho$ be positive real numbers such that $0<\lambda<1,0<\lambda\mu^{\rho}<1$. We set, for every real number $x$,

[TABLE]

with the convention that a sum indexed by an empty set equals zero.

Notice that $\Psi_{\rho}(\lambda,\mu)$ is a right continuous function in the variable $\rho$, while the function $\Phi_{\rho}(\lambda,\mu,x)$ is left continuous in both variables $\rho$ and $x$. Viewed as power series in the two variables $\lambda$ and $\mu$, these two functions are called Hecke-Mahler series which have been studied especially from a diophantine point of view [1, 2, 5, 10, 15, 17, 18, 20, 21]. We relate our functions $\delta(\lambda,\mu,\rho)$ and $\phi_{\lambda,\mu,\delta,\rho}(x)$ respectively to $\Psi_{\rho}(\lambda,\mu)$ and $\Phi_{\rho}(\lambda,\mu,x)$.

Lemma 2.1**.**

Let $\lambda,\mu,\rho$ be real numbers with $0<\lambda<1,\mu>0$ and $0<\rho<r_{\lambda,\mu}$, then the following equality holds

[TABLE]

Proof.

Reverting the summation order for the indices $h,k$ involved in $\Psi_{\rho}(\lambda,\mu)$, we obtain

[TABLE]

A positive integer $k$ is of the form $\left\lceil{h\over\rho}\right\rceil$ for some some positive integer $h$ if and only if $k-1<{h\over\rho}\leq k$, or equivalently $(k-1)\rho<h\leq k\rho$. There exists at most one integer $h$ in the interval $((k-1)\rho,k\rho]$ whose length is $\rho<r_{\lambda,\mu}<1$. The integer $h$ does exist exactly when $\lfloor(k-1)\rho\rfloor=\lfloor k\rho\rfloor-1$ and then $h=\lfloor k\rho\rfloor=\lfloor(k-1)\rho\rfloor+1$. Otherwise, $\lfloor(k-1)\rho\rfloor=\lfloor k\rho\rfloor$. Thus

[TABLE]

since $0<\rho<1$. ∎

We deduce

Corollary**.**

Let $0<\lambda<1$ and $\mu>0$, then the map $\rho\mapsto\delta(\lambda,\mu,\rho)$ is increasing on the interval $0<\rho<r_{\lambda,\mu}$ and sends the interval $(0,r_{\lambda,\mu})$ into the interval $(1-\lambda,d_{\lambda,\mu})$. Moreover, it is right continuous everywhere and continuous at any irrational point $\rho$.

Proof.

Using Lemma 2.1, we can rewrite $\delta(\lambda,\mu,\rho)$ in the form

[TABLE]

We distinguish two cases whether $\lambda\mu<1$ or not. When $\lambda\mu<1$, we have $d_{\lambda,\mu}=r_{\lambda,\mu}=1$. The series $\Psi_{\rho}(\lambda,\mu)$ converges for any $\rho\in(0,1)$ and the map $\rho\mapsto\Psi_{\rho}(\lambda,\mu)$ is obviously increasing, since $\Psi_{\rho}(\lambda,\mu)$ is a sum of powers of $\lambda$ and $\mu$ and that the set of summation indices $(h,k)$ enlarges when $\rho$ grows. We easily compute that

[TABLE]

It follows that

[TABLE]

for any $\rho\in(0,1)$. Since $\lambda$ differs from [math] and $1$, the homographic function

[TABLE]

is increasing on the interval $0<x<{\lambda^{2}\mu\over(1-\lambda)(1-\lambda\mu)}$, and sends this interval onto $(1-\lambda,1)$. By composition, we obtain that the image of $(0,1)$ by the map $\rho\mapsto\delta(\lambda,\mu,\rho)$ is contained in the interval $(1-\lambda,1)$. When $\lambda\mu\geq 1$, we have $r_{\lambda,\mu}=-\log\lambda/\log\mu$ and the series $\Psi_{\rho}(\lambda,\mu)$ tends to $+\infty$ when $\rho$ tends to $r_{\lambda,\mu}$ from below. Thus, we obtain in this case,

[TABLE]

For the continuity’s property, observe that the floor function $x\mapsto\lfloor x\rfloor$ is right continuous on ${\mathbb{R}}$ and continuous on ${\mathbb{R}}\setminus{\mathbb{Z}}$. ∎

We now give an alternative formula for the function $\phi_{\lambda,\mu,\delta,\rho}$ in terms of the Hecke-Mahler series $\Phi_{\rho}$.

Lemma 2.2**.**

Let $0<\lambda<1$, $\mu>0$ and $0<\rho<r_{\lambda,\mu}$, then, for any real number $x>-1$, we have the equalities

[TABLE]

where the indeterminate ratio $(1-1^{\lceil x\rceil})/(1-1)$ equals $\lceil x\rceil$ when $\mu=1$. Moreover, the formula

[TABLE]

holds for any real number $y$.

Proof.

From Definition 4, we can write

[TABLE]

which implies (5). For equation (6), multiplying (5) by $1-\lambda$, we find

[TABLE]

Observe that, for any integer $k\geq 0$, $\lceil(k+1)\rho+x\rceil-\lceil k\rho+x\rceil$ takes only the value 0 or 1. Therefore

[TABLE]

We obtain the equality

[TABLE]

from which formula (6) follows. The map $y\mapsto\lfloor y\rfloor-\lfloor y-k\rho\rfloor$ has period 1 for any integer $k$. We can thus replace $y$ by its fractional part $\{y\}$ in the sum over $k$ occurring in the definition 3 giving $\phi_{\lambda,\mu,\delta,\rho}$. Observe also that $\lfloor x\rfloor=-\lceil-x\rceil$ for any real number $x$. We can therefore rewrite $\phi_{\lambda,\mu,\delta,\rho}(y)$ in the form

[TABLE]

Using (5) and (6) for $x=-\{y\}$ and noting that $\lceil-\{y\}\rceil=0$, we obtain

[TABLE]

∎

Corollary**.**

*Let $0<\lambda<1$, $\mu>0$, $0<\rho<r_{\lambda,\mu}$ and $1-\lambda<\delta<d_{\lambda,\mu}$, then the function $\phi_{\lambda,\mu,\delta,\rho}$ is right continuous and non-decreasing on the interval $I=[0,1)$. Moreover,

(i) $\phi_{\lambda,\mu,\delta,\rho}$ is strictly increasing on $I$, if $\rho$ is irrational.

(ii) If $\rho={p\over q}$ is rational, the function $y\mapsto\phi_{\lambda,\mu,\delta,p/q}(y)$ is constant on each interval $[{n\over q},{n+1\over q}),\,n\in{\mathbb{Z}}$.

(iii) In any case, the relation $\phi_{\lambda,\mu,\delta,\rho}(y+1)=\phi_{\lambda,\mu,\delta,\rho}(y)+1$ holds for any real number $y$.*

Proof.

The function $x\mapsto\Phi_{\rho}(\lambda,\mu,x)$ is clearly non-decreasing and strictly increasing when $\rho$ is irrational. By Lemma 2.2, we have

[TABLE]

when $0\leq y<1$. Notice that, by the assumption, the coefficient $-{\delta-\mu(\lambda+\delta-1)\over\lambda}$ is negative which yields that $\phi_{\lambda,\mu,\delta,\rho}$ is non-decreasing. The other assertions are straightforward. ∎

2.2. Proof of Theorem 2

Let us begin with the following result on the transcendency of values of the Hecke-Mahler function, due to Loxton and Van der Poorten [17]. See also Sections 2.9 and 2.10 of the monograph [20] and the survey article [18].

Theorem 4**.**

Let $\lambda$ and $\mu$ be non-zero algebraic numbers and let $\rho$ be an irrational real number. Assume that $0<|\lambda|<1$ and $|\lambda||\mu|^{\rho}<1$. Then $\Psi_{\rho}(\lambda,\mu)$ is a transcendental number.

Using the homographic relations (4), both numbers $\delta(\lambda,\mu,\rho)$ and $\Psi_{\rho}(\lambda,\mu)$ are simultaneously either algebraic or transcendental. Then, it follows from Theorem 4 that $\delta(\lambda,\mu,\rho)$ is a transcendental number for any irrational real number $0<\rho<r_{\lambda,\mu}$. As a consequence of the assertion (ii) of Theorem 1, the rotation number $\rho_{\lambda,\mu,\delta}$ cannot be an irrational number $\rho$ when $\lambda,\mu,\delta$ are algebraic numbers. It is therefore a rational number. Theorem 2 is established.

3. Properties of the function $\phi$

Let $\lambda,\mu,\delta,\rho$ be four real numbers satisfying the inequalities

[TABLE]

We estimate in this technical section the value of the function $\phi_{\lambda,\mu,\delta,\rho}$ at the points [math] and $1-\rho$ according to the values of $\delta$. We stress that $\rho$ is not assumed here to be the rotation number of the map $f_{\lambda,\mu,\delta}$. On the opposite, we shall make use of our results to identify this rotation number $\rho_{\lambda,\mu,\delta}$ in the subsequent Section 5, and thus proving Theorem 1. Our estimates are based on numerical relations betweeen some special values of the Hecke-Mahler series $\Phi_{\rho}$ and the function $\sigma$, as for instance the formulae (9) to (12) below.

Lemma 3.1**.**

Assume that $0<\rho<r_{\lambda,\mu}$ is irrational. Let $\delta=\delta(\lambda,\mu,\rho)$ and $\phi=\phi_{\lambda,\mu,\delta,\rho}$. Then the following equalities hold

[TABLE]

Proof.

Recall the formula

[TABLE]

Notice first that we have the equalities

[TABLE]

the last one coming from Lemma 2.1 and noting that the strict inequality $l<k\rho$ is equivalent to $l\leq k\rho$, when $\rho$ is irrational. It follows from Lemma 2.2 and (7) that

[TABLE]

For the value $\phi(1-\rho)$, we compute $\Phi_{\rho}(\lambda,\mu,\rho-1)$ using (6). Noting that $\lceil\rho-1\rceil=0$, we find

[TABLE]

since $\lceil k\rho\rceil=\lfloor k\rho\rfloor+1$ for any integer $k\geq 1$. Therefore

[TABLE]

since $\delta=(1-\lambda)(1+\mu\sigma)/(1+(\mu-1)\sigma)$. ∎

When $\rho$ is a rational number $p/q$, the function $\phi_{\lambda,\mu,\delta,p/q}$ is constant on any interval of the form $[{n\over q},{n+1\over q}),n\in{\mathbb{Z}}$, and has a positive jump at the endpoints ${\mathbb{Z}}/q$. In this case, we have the analogous

Lemma 3.2**.**

Assume that $\rho=p/q$ and $\delta\left(\lambda,\mu,\left(p/q\right)^{-}\right)\leq\delta<\delta(\lambda,\mu,p/q).$ Put $\phi=\phi_{\lambda,\mu,\delta,p/q}$. Then

[TABLE]

and

[TABLE]

Proof.

Set $\sigma=\sigma(\lambda,\mu,{p\over q})$ and $\sigma^{-}=\sigma(\lambda,\mu,({p\over q})^{-})$. We first show that

[TABLE]

where we recall the notation

[TABLE]

from Theorem 1. By Definition 2, we have

[TABLE]

Observe that $\lfloor(k+q){p\over q}\rfloor=\lfloor k{p\over q}\rfloor+p$. Splitting the above sum over $k$ according to the various classes of $k$ modulo $q$, we obtain the first formula

[TABLE]

Similarly, we have

[TABLE]

Now, we establish the formulae

[TABLE]

To that purpose, we observe that the function $x\mapsto\Phi_{p/q}(\lambda,\mu,x)$ is constant on each interval $({n\over q},{n+1\over q}],n\in{\mathbb{Z}}$, and we use formula (6). Gathering as above the various classes of $k$ modulo $q$, we obtain the sums

[TABLE]

since $\left\lceil k{p\over q}\right\rceil=\left\lfloor k{p\over q}\right\rfloor+1$ for $1\leq k\leq q-1$ and $\left\lceil 0{p\over q}\right\rceil=0$. Similarly, we have the equalities

[TABLE]

since $\left\lceil{kp+1\over q}\right\rceil=\left\lceil{kp\over q}\right\rceil=\left\lfloor{kp\over q}\right\rfloor+1$ for $1\leq k\leq q-1$. For the value $x={p\over q}-1$, we find

[TABLE]

taking again the computations used for $\Phi_{p/q}(\lambda,\mu,0)$. Finally, we get

[TABLE]

by the above computation of $\Phi_{p/q}(\lambda,\mu,{1\over q}-1)$. The formulae (9) to (12) are established. We now use Lemma 2.2 in order to estimate values of $\phi$. We have

[TABLE]

Then (9) yields

[TABLE]

since $\delta(\lambda,\mu,(p/q)^{-})=(1-\lambda)(1+\mu\sigma^{-})/(1+(\mu-1)\sigma^{-})$. Observe now that the factor $1+(\mu-1)\sigma^{-}$ is always positive. Indeed, we deduce from Lemma 2.1 and its corollary that

[TABLE]

Therefore $1+(\mu-1)\sigma^{-}$ is bounded from below by $1$ when $\mu\geq 1$ and by $(1-\lambda)/(1-\lambda\mu)$ when $\mu<1$. It follows that $\phi(0)$ is $\geq 0$ if and only if $\delta\geq\delta(\lambda,\mu,(p/q)^{-})$. For the value $\phi(0^{-})$, observe that $-\{y\}=-1-y$ tends to $-1$ from above when $y<0$ tends to 0. Lemma 2.2 provides now the formula

[TABLE]

Then (10) yields

[TABLE]

It follows that $\phi(0^{-})$ is negative if and only if $\delta<\delta\left(\lambda,\mu,p/q\right)$. We now deal with the lower bound at the point $1-p/q$. Using Lemma 2.2 and (11), we find

[TABLE]

since the expression

[TABLE]

appearing above in the numerator is $\geq 0$. The computations are similar for the left limit at the point $1-p/q$. Using (12), we find

[TABLE]

since

[TABLE]

∎

4. The lift $F$

Let $F:{\mathbb{R}}\mapsto{\mathbb{R}}$ be the real function defined by

[TABLE]

Then $F$ is a lift of $f$, meaning that $F$ satisfies the following properties:

(i) For every $x\in{\mathbb{R}}$, we have

[TABLE]

(ii) $F(x+1)=F(x)+1$, for every $x\in{\mathbb{R}}$.

(iii) $F$ is an increasing function on ${\mathbb{R}}$ which is continuous on each interval of ${\mathbb{R}}\setminus{\mathbb{Z}}$ and right continuous everywhere.

Let $x\in{\mathbb{R}}$ and let $(x_{k})_{k\geq 0}$ be the forward orbit of $x$ by $F$, where $x_{k}=F^{k}(x)$ stands for the $k$-th iterate of the function $F$. When $\{x\}$ belongs to $C=\cap_{k\geq 1}f^{k}(I)$, we denote moreover by $(x_{-k})_{k\geq 0}$ the backward orbit of $x$ by $F$, where $x_{-k}=F^{-k}(x)$ is the $k$-th preimage of $x$ by $F$. This makes sense since $\{x_{-k}\}=f^{-k}(\{x\})$ is the $k$-th preimage of $\{x\}$ by $f$ and $x_{-k-1}=F^{-1}(x_{-k})$ is the inverse image of $x_{-k}$ by the injective map $F$, noting that $x_{-k}\in F({\mathbb{R}})=f(I)+{\mathbb{Z}}$ for all $k\geq 0$. A fundamental property is that any forward orbit $(x_{k})_{k\geq 0}$ can be computed explicitely in terms of its initial point $x$ and of the associated *symbolic sequence *

[TABLE]

It turns out that, for any orbit, this symbolic sequence is either periodic when the rotation number $\rho$ is rational, or a sturmian sequence of slope $\rho$ in the irrational case. We have the following explicit recursion formulae which motivate our definition of the conjugation $\phi$:

Lemma 4.1**.**

Let $x\in{\mathbb{R}}$ and let $(x_{k})_{k\geq 0}$ be the forward orbit of $x$ by $F$. For any non-negative integers $l$ and $n$, we have the relation

[TABLE]

Moreover, assume that $\{x\}\in C$. Let $(x_{-k})_{k\geq 0}$ be the backward orbit of $x$ by $F$ and assume that there exist two real numbers $y$ and $\rho$ with $0<\rho<1$ such that $\lfloor x_{k}\rfloor=\lfloor y+k\rho\rfloor$ for all integer $k\leq 0$. Then, we have the series expansion

[TABLE]

Proof.

Notice that $\lfloor x_{k+1}\rfloor-\lfloor x_{k}\rfloor\in\{0,1\}$ for any integer $k$. Put

[TABLE]

so that $F(x_{k})=a_{k}x_{k}+b_{k}$. Thus $x_{k+1}=a_{k}x_{k}+b_{k}$. Let $n$ be a positive integer. Composing these affine relations for $k=l,\dots,l+n-1$, we obtain by induction on $n$ the formula

[TABLE]

where we have set

[TABLE]

We display the terms

[TABLE]

appearing in the above sum. Note that $v_{n}=1$. Thus, by Abel’s summation, we find

[TABLE]

Replacing the index of summation $k$ by $n-k$ in the last sum, we find

[TABLE]

The first assertion is established. For the second one, observe that the first assertion remains valid for $l$ negative when $\{x\}$ belongs to $C$, since then $x$ has a preimage $F^{l}(x)$ and we apply the formula at this point. Choosing $l=-n$ and letting $n$ tend to infinity, we get the stated series expansion. ∎

The next result is crucial in our approach. It shows that $\phi$ satisfies a functional equation as in Theorem 3.

Lemma 4.2**.**

Let $\lambda$, $\mu$, $\delta$ and $\rho$ be real numbers such that $0<\lambda<1,\mu>0$, $0<\rho<r_{\lambda,\mu}$. We assume that $\delta=\delta(\lambda,\mu,\rho)$ when $\rho$ is irrational, or that $\delta$ belongs to the interval (2) when $\rho=p/q$ is rational. Put $\phi=\phi_{\lambda,\mu,\delta,\rho}$ and $F=F_{\lambda,\mu,\delta}$. Then, the relations

[TABLE]

hold for any real number $y$. Thus, the $F$-orbit $(x_{k})_{k\in{\mathbb{Z}}}$ of $x=\phi(y)$ is given by the sequence

[TABLE]

Proof.

We first show that $0\leq\phi(y)<1$ when $0\leq y<1$. In the case $\rho$ irrational, Lemma 3.1 gives $\phi(0)=0$. Thus $\phi(1)=\phi(0)+1=1$ and the corollary of Lemma 2.2 asserts that the function $\phi$ is strictly increasing. Therefore

[TABLE]

In the rational case, the function $\phi$ is non-increasing and constant on each interval $[{n\over q},{n+1\over q}),n\in{\mathbb{Z}}$. Now, we know that $\phi(0)\geq 0$ and $\phi(0^{-})<0$ by Lemma 3.2. Therefore

[TABLE]

For any $y\in{\mathbb{R}}$, we can write

[TABLE]

We have thus proved that $\lfloor\phi(y)\rfloor=\lfloor y\rfloor$ and $\{\phi(y)\}=\phi(\{y\})$ for all real number $y$. We now prove the relation $F(\phi(y))=\phi(y+\rho)$. By definition of $F$, we have to deal with two expressions for the value of $F(\phi(y))$ depending whether the fractional part $\{\phi(y)\}$ is smaller than $\eta=(1-\delta)/\lambda$ or not. But $\{\phi(y)\}=\phi(\{y\})$ and Lemmae 3.1 and 3.2 yield that $\phi(\{y\})$ belongs the interval $[0,\eta)$ when $\{y\}<1-\rho$, and to the other interval $[\eta,1)$ when $\{y\}\geq 1-\rho$, since $\phi$ is non-decreasing. The computation splits into two cases. Suppose first that $\{y\}<1-\rho$. Then $\lfloor y+\rho\rfloor=\lfloor y\rfloor$. Moreover, $\{\phi(y)\}=\phi(\{y\})<\eta$ by Lemmae 3.1 and 3.2 and the increasing monotonicity of the function $\phi$. Using the expression of $F$ in the intervals $[n,n+\eta),n\in{\mathbb{Z}}$, we obtain the equalities:

[TABLE]

The case $\{y\}\geq 1-\rho$ is similar. Then $\lfloor y+\rho\rfloor=\lfloor y\rfloor+1$ and $\{\phi(y)\}\geq\eta$ by Lemmae 3.1 and 3.2. We now use the expression of $F$ in the intervals $[n+\eta,n+1),n\in{\mathbb{Z}}$. We then obtain the equalities:

[TABLE]

∎

5. Proof of Theorem 1

Let $x\in{\mathbb{R}}$ and let $(x_{k})_{k\geq 0}$ be the forward orbit of $x$ by $F$. It is known (see [24]) that the limit

[TABLE]

exists and does not depend on the initial point $x$. The number $\rho_{\lambda,\mu,\delta}$ is called the rotation number of the map $f=f_{\lambda,\mu,\delta}$. Fix $\lambda$ and $\mu$ with $0<\lambda<1$, $\mu>0$. Let $\delta$ be a real number in the interval $1-\lambda<\delta<d_{\lambda,\mu}$. By the corollary of Lemma 2.1, the following alternative holds. Either $\delta$ belongs to the image of the interval $0<\rho<r_{\lambda,\mu}$ by the function $\rho\mapsto\delta(\lambda,\mu,\rho)$, or

[TABLE]

for some rational number $p/q$ with $0<p/q<1$ (these intervals are the jumps of the increasing function $\rho\mapsto\delta(\lambda,\mu,\rho)$). In the latter case, Lemma 4.2 yields that $\rho_{\lambda,\mu,\delta}=p/q$. Indeed, we select an initial point $x$ of the form $x=\phi_{\lambda,\mu,\delta,p/q}(y)$ for an arbitrary $y\in{\mathbb{R}}$, so that $x_{k}=\phi_{\lambda,\mu,\delta,p/q}(y+kp/q)$ for every integer $k\geq 0$. Then,

[TABLE]

It remains to deal with parameters $\delta$ in the image, in other words $\delta=\delta(\lambda,\mu,\rho)$ for some $0<\rho<r_{\lambda,\mu}$. When $\rho$ is irrational, Lemma 4.2 yields as well that $\rho_{\lambda,\mu,\delta}=\rho$. When $\rho=p/q$ is rational, we may use a general argument of continuity. Proposition 5.7 in [25] tells us that the rotation number $\rho_{\lambda,\mu,\delta}$ is a continuous function of the parameter $\delta$. Thus

[TABLE]

since we have already proved that $\rho_{\lambda,\mu,\delta^{\prime}}$ is constant and equal to $p/q$ when $\delta^{\prime}$ is located in the right open interval

[TABLE]

We express now $\delta(\lambda,\mu,p/q)$ and $\delta\left(\lambda,\mu,(p/q)^{-}\right)$ in term of the finite sum $S$. Recalling formula (8), we obtain

[TABLE]

and

[TABLE]

6. Irrational rotation number

We prove part (i) of Theorem 3 and we give furthermore a description of the iterated images $f^{n}(I)$ when the rotation number $\rho$ is irrational. In this case, the function $\phi$ is strictly increasing on ${\mathbb{R}}$ and jumps at the points $l\rho+{\mathbb{Z}},l\geq 1$. Put

[TABLE]

All the intervals $[\xi_{l}^{-},\xi_{l}),l\geq 1$, are pairwise disjoint and contained in $I=(0,1)$.

Proposition 5**.**

For any integer $n\geq 1$, we have the decomposition into disjoint intervals

[TABLE]

and the formulae

[TABLE]

Moreover, the set equalities

[TABLE]

hold. The set $\overline{C}$ is topologically homeomorphic to a Cantor set.

Proof.

Lemma 4.2 shows that for any $k\geq 0$ and any $l\geq 1$, we have the equalities

[TABLE]

and

[TABLE]

where $F^{k}$ stands for the $k$-th iterate of $F$. Since $F$ is increasing and continuous on ${\mathbb{R}}\setminus{\mathbb{Z}}$, it follows that

[TABLE]

so that any number $z\in F^{k}([\xi_{l}^{-},\xi_{l}))$ has integer part $\lfloor z\rfloor=\lfloor\{l\rho\}+k\rho\rfloor$. We first show that

[TABLE]

Since $\phi$ is right continuous and increasing, no point of $\phi(I)$ is located in an interval of the form $[\xi_{l}^{-},\xi_{l}),l\geq 1$. Thus, we have the inclusion $\phi(I)\subseteq I\setminus\bigcup_{l\geq 1}[\xi_{l}^{-},\xi_{l}).$ The reversed inclusion $\phi(I)\supseteq I\setminus\bigcup_{l\geq 1}[\xi_{l}^{-},\xi_{l})$ follows straightforwardly from the right continuity of $\phi$. Indeed, let $x\in I$ which is located outside the intervals $[\xi_{l}^{-},\xi_{l}),l\geq 1$. For every $n\geq 1$, define an index $l_{n}$ among the integers $1\leq l\leq n$ for which $x\leq\xi_{l}$ and $\xi_{l_{n}}$ is the closest to $x$. It is readily seen that the decreasing sequence $\{l_{n}\rho\}$ converges to a number $y$ and that $x=\phi(y)$ by right continuity of $\phi$. We know by Lemma 3.1 that the critical point $\eta=\phi(1-\rho)$ is located in the image $\phi(I)$. In particular, this critical point $\eta$ does not belong to any interval $[\xi_{l}^{-},\xi_{l}),l\geq 1$. The function $f$ is thus continuous on each interval $[\xi_{l}^{-},\xi_{l})$, so that we deduce from (15) that

[TABLE]

by reducing modulo 1. Now, Lemmae 3.1 and 4.2 yield the equalities

[TABLE]

Looking at Figure 1, we immediately observe that $f(I)=I\setminus[\xi_{1}^{-},\xi_{1})$, as announced. Taking now the image by $f$ and using (16), we find

[TABLE]

Arguing by induction on $n\geq 1$, we thus deduce from (16) the required equality

[TABLE]

Letting $n$ tend to infinity, we finally obtain that

[TABLE]

It remains to establish the explicit formulae giving $\xi_{l}$ and $\xi_{l}^{-}$. Lemma 4.1 delivers the expression

[TABLE]

since $\lfloor\phi(k\rho)\rfloor=\lfloor k\rho\rfloor$ for every integer $k$. Taking the fractional part, we obtain the formula

[TABLE]

The equality $\xi_{l}=f^{l}(0)$ immediately follows from Lemmae 3.1 and 4.2. Similarly $f^{l}(1^{-})=\xi_{l}^{-}=\{\phi(l\rho^{-})\}$. We have $\{0^{-}\}=1,\lfloor 0^{-}\rfloor=-1$ and $\lfloor k\rho^{-}\rfloor=\lfloor k\rho\rfloor$ for any integer $k\geq 1$. Then, Lemma 4.1 gives

[TABLE]

As a corollary of the above formulae, let us briefly prove that $\overline{C}$ is a Cantor set. The Lebesgue measure of $\overline{C}=I\setminus\bigcup_{l\geq 1}(\xi_{l}^{-},\xi_{l})$ is equal to

[TABLE]

Using (5) and (7), we easily compute the sum

[TABLE]

where $\sigma=\sigma(\lambda,\mu,\rho)$. On the other hand, we have

[TABLE]

Therefore $\overline{C}$ is a null set. Consequently, it has no inner point. Moreover, $\overline{C}=\overline{\phi(I)}$ has no isolated point, since the function $\phi$ is strictly increasing and right continuous. It follows that the compact set $\overline{C}$ is homeomorphic to a Cantor set. ∎

In order to complete the proof of the assertion (i) of Theorem 3, we now show that for any point $x\in I$, the $\omega$-limit set $\omega(x)$ coincides with $\overline{\phi(I)}$. To that purpose, we consider the $F$ orbit $x_{k}:=F^{k}(x),k\geq 0$, of $x$. When $x=\phi(y)$ belongs to $\phi(I)$, Lemma 4.2 shows that $x_{k}=\phi(y+k\rho)$ so that $f^{k}(x)=\{x_{k}\}=\phi(\{y+k\rho\})$. The sequence of fractional parts $(\{y+k\rho\})_{k\geq 0}$ is dense in $I$, since $\rho$ is an irrational number. Thus the set $\omega(x)$ of accumulation points of the orbit $(f^{k}(x))_{k\geq 0}$ is equal to $\overline{\phi(I)}$. It remains to deal with points $x\in I$ not belonging to $\phi(I)$, it means $x\in[\xi_{l}^{-},\xi_{l})$ for some $l\geq 1$. Observe first that

[TABLE]

since $x_{k}\in F^{k}([\xi_{l}^{-},\xi_{l}))$. In particular the symbolic sequence $(\lfloor x_{k+1}\rfloor-\lfloor x_{k}\rfloor)_{k\geq 0}$ coincides with the sturmian sequence $(\lfloor y+(k+1)\rho\rfloor-\lfloor y+k\rho\rfloor)_{k\geq 0}$, where we have set $y=\{l\rho\}$. Then, Lemma 4.1, with $l=0$ and $n=k$, provides us the formula

[TABLE]

Thus, the $F$-orbit $(x_{k})_{k}$ converges exponentially fast to the $F$-orbit $(\phi(y+k\rho))_{k\geq 0}$ as $k$ tends to infinity. Reducing modulo 1, one obtains as well that $\omega(x)=\overline{\phi(I)}$.

7. Rational rotation number

We prove the statement (ii) of Theorem 3 and add a dynamical description of the iterated images of $f=f_{\lambda,\mu,\delta}$. We assume throughout this section that the inequalities (2) are fulfilled for some rational number $p/q$. Then, Theorem 1 asserts that the rotation number of $f$ equals $p/q$. Put $\phi=\phi_{\lambda,\mu,\delta,p/q}$ and set

[TABLE]

It is convenient to extend the sequence $(\zeta_{m})_{0\leq m\leq q-1}$ by $q$-periodicity setting $\zeta_{m}=\zeta_{r}$ for any integer $m\in{\mathbb{Z}}$, where $r$ is the remainder in the euclidean division of $m$ by $q$. Then, Lemma 4.2 yields the formula

[TABLE]

As $\phi$ is nondecreasing $\zeta_{0}\leq\cdots\leq\zeta_{q-1}$, we claim that these numbers are distinct. If not, there exists $m$ such that $\zeta_{m}=\zeta_{m+1}$. Iterating $f$ thus $\zeta_{m+kp}=\zeta_{m+kp+1}$, $\forall k\in{\mathbb{Z}}$, obtaining that the function $\phi$ is constant, in contradiction for instance with Lemma 3.2. Moreover, $\zeta_{q-1}=\phi(1^{-})=\phi(0^{-})+1<1$, again by Lemma 3.2. So, we have an increasing sequence

[TABLE]

in $I$. It follows that the set $\phi(I)=\{\zeta_{0},\dots,\zeta_{q-1}\}$ is an $f$-cycle of order $q$, on which $f$ acts by the substitution $m\mapsto m+p$ modulo $q$. Recall that $f(\eta)=0$ and $f(\eta^{-})=1$, where $\eta=(1-\delta)/\lambda$ is the critical point of the map $f$. Moreover, Lemma 3.2 shows that $\zeta_{q-p-1}<\eta\leq\zeta_{q-p}$. If $\eta$ does not belong to $\phi(I)$, we have strict inequalities $\zeta_{q-p-1}<\eta<\zeta_{q-p}$ and $f(\zeta_{q-p})=\zeta_{0}>0$. Otherwise $\eta=\zeta_{q-p}$ and $\zeta_{0}=f(\eta)=0$. The latter case occurs only when $\delta$ coincides with the left end point of the interval (2). Indeed (13) shows that $\phi(0)$ vanishes if and only if $\delta=\delta\left(\lambda,\mu,(p/q)^{-}\right)$. \marginsize2.5cm2.5cm1cm2cm

The next proposition provides a partition of the image $f^{n}(I)$, $n\geq 1$, into disjoint intervals. It is convenient to consider circular intervals (or circle arcs identifying $I$ with ${\mathbb{R}}/{\mathbb{Z}}$). For any $a,b$ both belonging to $I$, we set

[TABLE]

We write for instance $[\zeta_{q-1},\zeta_{0})=[0,\zeta_{0})\cup[\zeta_{q-1},1)$.

Proposition 6**.**

For any integer $l\geq 1$, the circular interval \big{[}f^{l}(1^{-}),f^{l}(0)\big{)} is contained in \big{[}\zeta_{lp-1},\zeta_{lp}\big{)}=\big{[}f^{l}(\zeta_{q-1}),f^{l}(\zeta_{0})\big{)}. We have $f^{l}(1^{-})<f^{l}(0)$ when $l$ is not divisible by $q$ and $f^{l}(1^{-})>f^{l}(0)$ when $l$ is a multiple of $q$. The decomposition into disjoint intervals

[TABLE]

and

[TABLE]

holds true. Moreover $f^{n}(I)$ has Lebesgue measure $\leq(\lambda^{q}\mu^{p})^{\lfloor n/q\rfloor}$.

Proof.

Let us consider the partition of $I$

[TABLE]

into disjoint circular intervals. The action of $f$ on these intervals is drawn in Figure 6. The inclusions

[TABLE]

hold for $1\leq l\leq q$ and we have equality

[TABLE]

for $1\leq l\leq q-1$. However, for $l=q$, we have a strict inclusion:

[TABLE]

Thus (17) holds for $n=1$. Applying $f$ to this decomposition, we observe the appearance of a second “hole” $[f^{2}(1^{-}),f^{2}(0))$ contained in the interval $[\zeta_{2p-1},\zeta_{2p})$, just over the first hole $[f(1^{-}),f(0))$ dotted on Figure 6. Iterating $f$ again, we obtain (17) for $1\leq n\leq q$ (note that the formulae (17) and (18) coincide for $n=q$). At the $q$-th iteration, each interval \big{[}\zeta_{lp-1},\zeta_{lp}\big{)},1\leq l\leq q, has been holed and $\eta$ does not belong to $f^{q}(I)$. Then, $f$ exchanges the intervals and contracts them. We thus obtain formula (18) for $n\geq q$. We finally prove the bound $|f^{n}(I)|\leq(\lambda^{q}\mu^{p})^{\lfloor n/q\rfloor}$ for the Lebesgue measure of the iterated images $f^{n}(I)$. We claim that

[TABLE]

for every integer $n\geq 0$. Suppose for instance $n\geq q$ and rewrite (18) in the form

[TABLE]

Each interval $H=[\zeta_{lp-1},f^{l}(1^{-})$ or $H=[f^{l}(0),\zeta_{lp})$ involved in the disjoint union (20) is contained either in $[0,\eta)$ or in $[\eta,1)$. Keeping track of the iterated images

[TABLE]

for $1\leq m\leq q$ in the decomposition (20) at level $n+m$ (instead of $n$), we observe that $f^{m}(H)$ is an interval contained in $[0,\eta)$ for $q-p$ values of $m$ and in $[\eta,1)$ for the $p$ remaining values of $m$ (see Figure 5). Notice that the image $f(J)$ of any interval $J$ contained in $[0,\eta)$ (resp. $[\eta,1)$) is an interval whose lenght equals $\lambda|J|$ (resp. $\lambda\mu|J|$). The length of $f^{m+1}(H)$ equals the length of $f^{m}(H)$ multiplied either by $\lambda$ or by $\lambda\mu$, according whether $f^{m}(H)$ is contained in $[0,\eta)$ or in $[\eta,1)$. It follows that

[TABLE]

Summing over the disjoint intervals $H$ occurring in (20), we obtain (19) for $n\geq q$. The proof for $n\leq q-1$ is similar, now based on the decomposition (17). Using euclidean division, write $n=q\lfloor n/q\rfloor+r$, with $0\leq r<q$. Equation (19) yields the required bound

[TABLE]

∎

We deduce from Proposition 6 the following explicit decomposition of the images $f^{n}(I)$.

Corollary**.**

Let $n\geq q$. Denote by $\bar{p}$ the multiplicative inverse of $p$ modulo $q$. For every integer $l$ with $n-q+1\leq l\leq n$, let $m$ be the unique integer in the interval $n-q+1\leq m\leq n$ which is congruent to $l+\bar{p}$ modulo $q$. Then, the decomposition into disjoint intervals

[TABLE]

holds true. For every integer $l$ with $n-q+1\leq l\leq n$, the interval $[f^{l}(0),f^{m}(1^{-}))$ contains the point $\zeta_{lp}$.

Proof.

Recall the decomposition (20) and observe that $\zeta_{mp-1}=\zeta_{lp}$ and that $l\mapsto m$ is a bijection of the set $\{n-q+1,\ldots,n\}$. Collecting the intervals involved in (20) by pairs $(l,m)$, we find

[TABLE]

∎

It follows from the corollary that

[TABLE]

Indeed, $\{\zeta_{0},\dots,\zeta_{q-1}\}$ is contained in $f^{n}(I)$, for every $n\geq 1$. The image $f^{n}(I)$, for $n\geq q$, equals the union of $q$ intervals whose lengths shrink to [math], as $n\to\infty$, noting that Proposition 6 delivers the bound

[TABLE]

where in both upper bounds, the numbers $\max(\lambda,\lambda\mu)$ and $\mu^{p/q-r_{\lambda,\mu}}$ are less than $1$. This proves (21). Since these $q$ disjoint intervals rotates under the action of $f$, we obtain as well that the the $\omega$-limit set $\omega(x)$ equals $C$, for any $x\in I$. The proof of the assertion (ii) of Theorem 3 is complete.

8. The right end point

We deal here with the exceptional value $\delta=\delta(\lambda,\mu,p/q)$. Let us first explain the reasons why the arguments expanded in Section 7 do not apply to this value. Indeed, (14) shows that $\phi(0^{-})=0$ in this case. Then

[TABLE]

and the set $\{\zeta_{0},\dots,\zeta_{q-1}\}$ cannot be of course an $f$-cycle, since it contains the point $1$ which is outside the set of definition $I$ of the map $f$. We slightly modify the map $f$ in order that the obstruction no longer holds. Put $J=(0,1]$. As usual, let $\lambda,\mu,\delta$ be three real numbers with

[TABLE]

and let $\rho=\rho_{\lambda,\mu,\delta}$ be the rotation number of $f=f_{\lambda,\mu,\delta}$. Recall the associated lift $F=F_{\lambda,\mu,\delta}$ and the conjugation $\phi=\phi_{\lambda,\mu,\delta,\rho}$. We introduce three functions $f^{-},F^{-}$ and $\phi^{-}$ which are the left limit of $f,F$ and $\phi$ respectively.

Definition 5**.**

(i) Let $f^{-}:J\mapsto J$ be the map defined by

[TABLE]

(ii). Let $F^{-}:{\mathbb{R}}\mapsto{\mathbb{R}}$ be the map defined by

[TABLE]

(iii) Let $\phi^{-}:{\mathbb{R}}\mapsto{\mathbb{R}}$ be the map defined by

[TABLE]

The maps $f$ and $f^{-}$ share almost the same dynamical behaviour and we present the analogies, omitting the proofs which follow the lines of Sections 3 and 4. The two maps $f$ and $f^{-}$ coincide on $(0,1)\setminus\{\eta\}$ and differ at the critical point $\eta=(1-\delta)/\lambda$ where $f(\eta)=0$ and $f^{-}(\eta)=1$. Thus, any $f$-orbit contained in $(0,1)$ does not contain the point $\eta$ and is also an $f^{-}$-orbit. The function $F^{-}$ turns out to be a lift for the circle map $f^{-}$, identifying now the circle ${\mathbb{R}}/{\mathbb{Z}}$ with the interval $J$. It follows that both maps $f$ and $f^{-}$ have the same rotation number $\rho=\rho_{\lambda,\mu,\delta}$. One can show that when $\rho$ is irrational, or when $\rho=p/q$ is rational and

[TABLE]

the functional equation $F^{-}(\phi^{-}(y))=\phi^{-}(y+\rho)$ holds for any $y\in{\mathbb{R}}$. Moreover, when $\rho$ is irrational, the closure $\overline{\phi^{-}(J)}=\overline{\phi(I)}$ is the Cantor set considered in Theorem 3 (i) and any $f^{-}$-orbit approaches of this Cantor set. When (22) holds, every $f^{-}$-orbit approaches cyclically of the periodic $f^{-}$-cycle

[TABLE]

of order $q$. From now, let us fix $\delta=\delta(\lambda,\mu,p/q)$ and put $f=f_{\lambda,\mu,\delta}$. Since $\delta$ belongs to the interval (22), $\phi(I)$ is an $f^{-}$-cycle of order $q$ containing the point $1$. Let ${\mathcal{O}}$ be an $f$-orbit. If $0\notin{\mathcal{O}}$, then ${\mathcal{O}}$ is also an $f^{-}$-orbit which converges to $\phi(I)$. It follows in particular that there exists no finite $f$-cycle. Indeed, arguing as in Section 7, this finite cycle should be an attractor for any $f$-orbit ${\mathcal{O}}$, and it would be equal to $\phi(I)$, which is impossible since $1\in\phi(I)$. Suppose now that $0\in{\mathcal{O}}$. Then [math] appears only once in ${\mathcal{O}}$. If not, ${\mathcal{O}}$ would contain a finite $f$-cycle. Hence, some tail of ${\mathcal{O}}$ does not contain [math] and ${\mathcal{O}}$ converges as well to $\phi(I)$. Part (iii) of Theorem 3 is established. Acknowledgements. We graciously acknowledge the support of Région Provence-Alpes-Côte d’Azur through the project APEX Systèmes dynamiques: Probabilités et Approximation Diophantienne PAD, CEFIPRA through the project No. 5801-B and the program MATHAMSUD projet No. 38889TM DCS: Dynamics of Cantor systems.