The entropy of alpha-continued fractions: analytical results

TL;DR

This paper investigates the ergodic properties of a family of interval maps related to generalized continued fractions, establishing the Hölder continuity of entropy and proving a central limit theorem for certain observables.

Contribution

It provides new analytical results on the entropy dependence and statistical behavior of a family of continued fraction maps, expanding understanding of their ergodic theory.

Findings

Entropy varies Hölder-continuously with the parameter alpha

A central limit theorem is established for a broad class of observables

Results apply to Birkhoff averages converging to the entropy

Abstract

We study the ergodic theory of a one-parameter family of interval maps T_alpha arising from generalized continued fraction algorithms. First of all, we prove the dependence of the metric entropy of T_alpha to be Hoelder-continuous in the parameter alpha. Moreover, we prove a central limit theorem for possibly unbounded observables whose bounded variation grows moderately. This class of functions is large enough to cover the case of Birkhoff averages converging to the entropy.

Equations

$$T_{\alpha }(x)=\frac{1}{|x|}-a_{\alpha }(x)$$

$$x=a_{0,\alpha }+\frac{{ϵ}_{1,\alpha }}{a_{1,\alpha }+\frac{{ϵ}_{2,\alpha }}{a_{2,\alpha }{+}_{\ddots }}}$$

$$h(T_{\alpha })={\int }_{\alpha -1}^{\alpha }\log |T_{\alpha }^{\prime }|d{\mu }_{\alpha }$$

$$\underset{n\to \infty }{\lim }{\mu }_{\alpha }\left(\frac{S_n(f-{\int }_{I_{\alpha }}fd{\mu }_{\alpha })}{\sqrt{n}}\le v\right)=\frac{1}{\sigma \sqrt{2\pi }}{\int }_v^{+\infty }{e}^{-\frac{t^2}{2{\sigma }^2}}dt\forall v\in \mathbb{R}$$

$$\underset{X}{\mathrm{Var}}f:=\sup \sum _{i=1}^n|f(x_i)-f(x_{i+1})|$$

$$\Vert f{\Vert }_{BV(I)}:=\inf \left\{\underset{I}{\mathrm{Var}}g+{\int }_I|g(x)|dx:g=fa.e.\right\}$$

$$f(x)\in [\underset{y\to x^{-}}{\lim }f(y),\underset{y\to x^{+}}{\lim }f(y)]\forall x\in I$$

$$I_j^{+}=\left(\frac{1}{j+\alpha },\frac{1}{j-1+\alpha }\right)\text{ if }j\ge j_{min}+1I_{j_{min}}^{+}=\left(\frac{1}{j_{min}+\alpha },\alpha \right)$$

$$I_j^{-}=\left(-\frac{1}{j-1+\alpha },-\frac{1}{j+\alpha }\right)\text{ if }j\ge 3I_2^{-}=\left(\alpha -1,\frac{1}{2+\alpha }\right)$$

$$\{I_{j_1}^{{ϵ}_1}\cap T_{\alpha }^{-1}(I_{j_2}^{{ϵ}_2})\cap \cdots \cap T^{-(n-1)}(I_{j_n}^{{ϵ}_n})|I_{j_1}^{{ϵ}_1},\dots ,I_{j_n}^{{ϵ}_n}\in {\mathcal{P}}_1\}$$

$$g_{n,\alpha }(x):=\sum _{j\in {\mathcal{P}}_n}\frac{1}{|(T_{\alpha }^n{)}^{\prime }(x)|}{\chi }_{I_j}(x)$$

$$\Vert g_{n,\alpha }{\Vert }_{\infty }\le {\gamma }_{\alpha }^n$$

$$\underset{j\in {\mathcal{P}}_n}{\sup }\underset{x\in I_j}{\sup }\left|g_{n,\alpha }^{\prime }(x)\right|\le \frac{2}{1-{\gamma }_{\alpha }}$$

$$\#\{T_{\alpha }^n(I_j)|I_j\in {\mathcal{P}}_n\}\le 2n+1$$

$$\underset{I_{\alpha }}{\mathrm{Var}}{g}_{1,\alpha }\le C_0<+\infty \forall \alpha \in (0,1)$$

$${\int }_{I_{\alpha }}{\Phi }_{\alpha }(f)gdx={\int }_{I_{\alpha }}f(g\circ T_{\alpha })dx\forall f\in L^{\infty }(I_{\alpha })$$

$${\Phi }_{\alpha }^n(f)=\sum _{j\in {\mathcal{P}}_n}\frac{f\circ {\sigma }_j}{|(T_{\alpha }^n{)}^{\prime }\circ {\sigma }_j|}{\chi }_{T_{\alpha }^n(I_j)}$$

$${\Phi }_{\alpha }={\Pi }_{\alpha }+{\Psi }_{\alpha }$$

$${\Pi }_{\alpha }(f)=\underset{n\to \infty }{\lim }\frac{1}{n}\sum _{k=1}^n{\Phi }_{\alpha }^k(f)$$

$${\Phi }_{\alpha }=\sum _{i=0}^p{\lambda }_i{\Phi }_i+{\Psi }_{\alpha }$$

$$\left|{\int }_{I_{\alpha }}{f}_1(x)f_2(T_{\alpha }^n(x))d{\mu }_{\alpha }-{\int }_{I_{\alpha }}{f}_1(x)d{\mu }_{\alpha }{\int }_{I_{\alpha }}{f}_2(x)d{\mu }_{\alpha }\right|\le C{\lambda }^n\Vert f_1{\Vert }_{BV}\Vert f_2{\Vert }_{L^1}$$

$$\mathrm{Var}{\Phi }_{\alpha }^n(f)\le C{\lambda }^n\mathrm{Var}f+D\Vert f{\Vert }_{L^1}\forall n\ge 1,\forall f\in BV(I_{\alpha })$$

$$\mathrm{Var}{\Phi }_{\alpha }^n(f)\le \mathrm{Var}(f\cdot g_{n,\alpha })$$

$$\mathrm{Var}{\Phi }_{\alpha }^n(f)=\mathrm{Var}\left(\sum _{j\in {\mathcal{P}}_n}\frac{f\circ {\sigma }_j}{|(T_{\alpha }^n{)}^{\prime }\circ {\sigma }_j|}{\chi }_{T_{\alpha }^n(I_j)}\right)\le \sum _{j\in {\mathcal{P}}_n}\mathrm{Var}\left(\frac{f\circ {\sigma }_j}{|(T_{\alpha }^n{)}^{\prime }\circ {\sigma }_j|}{\chi }_{T_{\alpha }^n(I_j)}\right)=$$

$$=\sum _{j\in {\mathcal{P}}_n}\mathrm{Var}\left(\frac{f}{|(T_{\alpha }^n{)}^{\prime }|}{\chi }_{I_j}\right)=\mathrm{Var}\left(f\sum _{j\in {\mathcal{P}}_n}\frac{1}{|(T_{\alpha }^n{)}^{\prime }|}{\chi }_{I_j}\right)=\mathrm{Var}(f{g}_{n,\alpha })$$

$$g_{n,\alpha }=h_{n,\alpha }+l_{n,\alpha }$$

$$g_{1,\alpha}(x):=\left\{\begin{array}[]{ll}x^{2}&\textup{if }x\textup{ belongs to some }I_{j}\\ 0&\textup{otherwise}\par\end{array}\right.$$

$$l_{1,\alpha }:=g_{1,\alpha }{\chi }_L{h}_{1,\alpha }:=g_{1,\alpha }{\chi }_{I_{\alpha }\setminus L}$$

$$\begin{array}[]{lll}h_{n+1,\alpha}&:=&h_{n,\alpha}\circ T_{\alpha}\cdot h_{1,\alpha}\\ l_{n+1,\alpha}&:=&l_{n,\alpha}\circ T_{\alpha}\cdot g_{1,\alpha}+h_{n,\alpha}\circ T_{\alpha}\cdot l_{1,\alpha}\end{array}$$

$$\underset{I_{\alpha }}{\mathrm{Var}}{h}_{n,\alpha }\le 2^{n-1}{C}_0^n\forall \alpha \in (0,1),\forall n\ge 1$$

Full text

MSC: 11K50, 37A10 (Primary) 37A55, 37E05 (Secondary)

1. Introduction

Let $\alpha\in[0,1]$. Let us define the map $T_{\alpha}:[\alpha-1,\alpha]\rightarrow[\alpha-1,\alpha]$ as $T_{\alpha}(0)=0$ and

[TABLE]

with $a_{\alpha}(x):=\left\lfloor\frac{1}{|x|}+1-\alpha\right\rfloor$. These systems were introduced by Nakada [12] and are known in the literature as $\alpha$-continued fractions, or Japanese continued fractions. By taking $x_{n,\alpha}=T_{\alpha}^{n}(x)$, $a_{n,\alpha}=a_{\alpha}(x_{{n-1},\alpha})$, $\epsilon_{n,\alpha}=\textup{Sign}(x_{n-1,\alpha})$, the orbit under $T_{\alpha}$ generates the generalized continued fraction expansion

[TABLE]

The algorithm, analogously to the Gauss map in the classical case, provides rational approximations of real numbers. It is known that for each $\alpha\in(0,1]$ there exists a unique invariant measure $\mu_{\alpha}(dx)=\rho_{\alpha}(x)dx$ absolutely continuous w.r.t. Lebesgue measure, and this measure is ergodic (see [10]). In this paper we will focus on the metric entropy of the $T_{\alpha}$’s, which is given by Rohlin’s formula (see [14])

[TABLE]

Nakada [12] computed exact values of $h(T_{\alpha})$ for $\alpha\geq\frac{1}{2}$, showing that in this interval $h(T_{\alpha})$ is continuous, and smooth except for the point $\alpha=\frac{\sqrt{5}-1}{2}$, where the left and right derivatives do not coincide.

In [10], Luzzi and Marmi studied the behaviour of $h(T_{\alpha})$ as a function of $\alpha$ for all parameters $\alpha\in(0,1)$. They gave numerical evidence that this function is continuous but not smooth. Nakada and Natsui [13] then proved that this function is indeed not monotone, by giving explicit construction of infinitely many monotonicity intervals. An extensive numerical study of these intervals has been carried out in [2], and a complete characterization of all monotonicity intervals is given in [3]. Even though the entropy is conjecturally smooth on any such interval, there are points at which it is not even locally monotone, and the set of bifurcation parameters has a complicated self-similar structure.

The first major result of this paper is the

Theorem 1.1**.**

The entropy function $\alpha\mapsto h(T_{\alpha})$ is Hölder-continuous of any exponent $0<s<\frac{1}{2}$.

The proof follows from spectral analysis of the transfer operator acting on the space of functions of bounded variation. First (section 3.1), we prove a uniform bound on the essential spectral radius (Lasota-Yorke inequality). Then (section 3.2), we prove that a suitable distance between the transformations $T_{\alpha}$ is Hölder-continuous in $\alpha$, and use a stability result of the spectral decomposition [7] to prove Hölder-continuity of the invariant densities $\rho_{\alpha}$ in $L^{1}$-norm. Note that invariant densities are not continuous in $BV$-norm (see remark Remark).

The second part of the paper deals with central limit theorems. In [10], entropy is computed by approximating it with Birkhoff averages for the observable $\log|T_{\alpha}^{\prime}|$, and numerical evidence is given ([10], figure 3) that Birkhoff sums for different orbits distribute normally around the average. We first show (section 4.1) that the methods of [1] can be used to prove a central limit theorem for observables of bounded variation.

Moreover, in section 4.2 we expand the class of observables we use in order to encompass unbounded observables such as the logarithm. Indeed, we will define a new family of Banach spaces $B_{K,\delta}$, consisting of possibly unbounded functions whose total variation grows slowly on intervals which approach zero. Such functions will be called of mild growth, and we prove the central limit theorem to hold in these larger spaces:

Theorem 1.2**.**

Let $\alpha\in(0,1]$, $0<\delta<\frac{1}{2}$, and $K$ sufficiently large. Then, for every non-constant real-valued $f\in B_{K,\delta}$ there exists $\sigma>0$ s.t.

[TABLE]

As a corollary, Birkhoff sums for the observable $\log|T_{\alpha}^{\prime}|$ distribute normally around the average value $h(T_{\alpha})$.

Finally, in section 5 we discuss the dependence of the standard deviation of Birkhoff averages on the parameter $\alpha$. More precisely, given some observable $f$ of class $C^{1}$, for which we proved the central limit theorem to hold, we will prove that the variance $\sigma^{2}_{\alpha,f}$ of the limit Gaussian distribution is continuous in $\alpha$. The result is motivated by numerical data in ([2], section 2.3).

Many different authors have studied the spectral properties of transfer operators of expanding maps. For instance, a spectral decomposition for individual expanding maps is proved in [1], [15] and [17]. A brief historical account with references is given in [7]. In our case, however, it is essential to prove estimates on the spectral radius which are uniform in $\alpha$. Since new branches of $T_{\alpha}$ appear as $\alpha$ moves, and $T_{\alpha}$ develops an indifferent fixed point as $\alpha\to 0$, proving uniformity requires more work. Unfortunately, although a uniform Lasota-Yorke inequality holds, the proof provided by [10] contains a bug; we shall therefore produce a new proof in prop. 3.1. Another proof of continuity (not Hölder) of entropy is given in the very recent paper [9] via a study of natural extensions.

Let us finally remark that our functional-analytic methods only use a few properties of $T_{\alpha}$, hence they can be applied to a wider class of one-parameter families of expanding interval maps. For instance, they apply to the case of $(a,b)$-continued fraction transformations studied in [8] for parameters on the critical line $b-a=1$.

2. Basic properties

Let us start by setting up the framework needed for the rest of the paper, and establishing a basic spectral decomposition for the transfer operator. The literature on thermodynamic formalism for interval maps is huge: the sources we mainly refer to are [1], [15] and [17], which already make use of functions of bounded variation.

The total variation of a function $f$ on a set $X\subseteq\mathbb{R}$ is

[TABLE]

where the sup is taken over all finite increasing sequences ${x_{1}\leq x_{2}\leq\dots\leq x_{n}}$ of points of $X$. Given an interval $I$, let us denote $BV(I)$ the Banach space of complex-valued bounded variation functions of the interval $I$, modulo equality almost everywhere. The space is endowed with the norm

[TABLE]

Observe that every $f\in BV(I)$ has a (not necessarily unique) representative of minimal total variation, namely such that

[TABLE]

In the following, we will always choose representatives for our functions of minimal variation. Other basic properties of total variation are stated in the appendix.

2.1. Cylinders

For each $\alpha\in(0,1)$, the dynamical system $T_{\alpha}$ defined in the introduction acts on the interval $I_{\alpha}:=[\alpha-1,\alpha]$. Observe that there exists a partition of $I_{\alpha}$ in a countable number of intervals $I_{j}$ such that for every $j$ the restriction $T_{\alpha}\mid_{I_{j}}$ is a strictly monotone, $C^{\infty}$ function and it extends to a $C^{\infty}$ function on the closure of every $I_{j}$. The least fine of such partitions will be called $\mathcal{P}_{1}$, the *partition associated * to $T_{\alpha}$. More specifically, $\mathcal{P}_{1}=\{I_{j}^{+}\}_{j\geq j_{min}}\cup\{I_{j}^{-}\}_{j\geq 2}$ with $j_{min}=\lceil\frac{1}{\alpha}-\alpha\rceil$ where

[TABLE]

[TABLE]

Moreover, for every $n>1$, the set

[TABLE]

where $\epsilon_{i}\in\{+,-\}$, is a partition of $I_{\alpha}$ in a countable number of intervals such that on each of these the restriction of $T_{\alpha}^{n}$ is monotone and $C^{\infty}$: such a partition will be denoted by $\mathcal{P}_{n}$ and its elements called cylinders. The cylinder $I_{j_{1}}^{\epsilon_{1}}\cap T_{\alpha}^{-1}(I_{j_{2}}^{\epsilon_{2}})\cap\dots\cap T^{-(n-1)}(I_{j_{n}}^{\epsilon_{n}})$ will be denoted either by $(I_{j_{1}}^{\epsilon_{1}},\dots,I_{j_{n}}^{\epsilon_{n}})$ or by $((j_{1},\epsilon_{1}),\dots,(j_{n},\epsilon_{n}))$. The cylinders $I_{j}\in\mathcal{P}_{n}$ such that $T_{\alpha}^{n}(I_{j})=I_{\alpha}$ will be called full cylinders.

Let us define the function

[TABLE]

The following estimates, proven in the appendix, will be used throughout the paper:

Proposition 2.1**.**

For every $\alpha\in(0,1)$ and for every $n\geq 1$

1. (1)

[TABLE]

where $\gamma_{\alpha}:=\max\{\alpha^{2},(\alpha-1)^{2}\}$. 2. (2)

[TABLE] 3. (3)

The set $\{T^{n}_{\alpha}(I_{j})\ |\ I_{j}\in\mathcal{P}_{n}\}$ is finite; more precisely,

[TABLE] 4. (4)

The total variation of $g_{1,\alpha}$ is universally bounded, i.e. there is a constant $C_{0}$ such that

[TABLE]

2.2. Spectral decomposition

The transfer operator (also known as Ruelle-Perron-Frobenius operator) $\Phi_{\alpha}:L^{1}(I_{\alpha})\rightarrow L^{1}(I_{\alpha})$ is defined via the duality

[TABLE]

Let us recall that the $n^{th}$ iterate of the transfer operator is given by

[TABLE]

where $\sigma_{j}:T^{n}_{\alpha}(I_{j})\rightarrow I_{j}$ is the inverse of the restriction $T^{n}_{\alpha}\mid_{I_{j}}:I_{j}\rightarrow T^{n}_{\alpha}(I_{j})$.

Even though $\Phi_{\alpha}$ is so far defined on $L^{1}$, it turns out that the transfer operator preserves the subspace $BV(I_{\alpha})$, and indeed it has good convergence properties in $BV$-norm. More precisely, we can now prove the

Theorem 2.2**.**

Let $\Phi_{\alpha}:L^{1}(I_{\alpha})\rightarrow L^{1}(I_{\alpha})$ be the transfer operator for the system $T_{\alpha}$, with $\alpha\in(0,1)$. Then one can write

[TABLE]

where $\Pi_{\alpha}$ and $\Psi_{\alpha}$ are commuting, linear bounded operators on $BV(I_{\alpha})$. Moreover, $\Psi_{\alpha}$ is a linear bounded operator on $BV(I_{\alpha})$ of spectral radius strictly less than $1$, and $\Pi_{\alpha}$ is a projector onto the one-dimensional eigenspace relative to the eigenvalue $1$. It is given by

[TABLE]

where the convergence is in $L^{1}$.

Corollary 2.3**.**

For every $\alpha\in(0,1)$, $T_{\alpha}$ has exactly one invariant probability measure $\mu_{\alpha}$ which is absolutely continuous w.r.t. Lebesgue measure. Its density will be denoted by $\rho_{\alpha}$.

Proof.

Let us fix $\alpha\in(0,1)$. By proposition 2.1, we can apply ([1], prop. 4.1), which yields via Ionescu-Tulcea and Marinescu’s theorem [5] the following spectral decomposition

[TABLE]

where $|\lambda_{i}|=1$, and the $\Phi_{i}$ are linear bounded operators on $BV(I_{\alpha})$ with finite-dimensional image, and $\rho(\Psi_{\alpha})<1$. Now, it is known ([10], lemma 1) that $T_{\alpha}$ is exact in Rohlin’s sense (see [14]); this implies that the invariant measure we have found is ergodic and mixing, which in turn implies that the only eigenvalue of $\Phi_{\alpha}$ of modulus $1$ is $1$ itself and that its associated eigenspace is one-dimensional (see [17], chap. 3). ∎

The spectral decomposition also immediately implies the following exponential decay of correlations:

Proposition 2.4**.**

For any $\alpha\in(0,1)$ there exist $C,\lambda$, $0<\lambda<1$ such that for every $n\in\mathbb{N}$ and for every $f_{1},f_{2}\in BV(I_{\alpha})$

[TABLE]

Proof.

One can take any $\lambda$ s.t. $\rho(\Psi_{\alpha})<\lambda<1$ and $C=2\|\rho_{\alpha}\|_{BV}\sup_{n\in\mathbb{N}}\frac{\|\Psi_{\alpha}^{n}\|_{BV}}{\lambda^{n}}$. ∎

3. Continuity of entropy

The goal of this section is to prove theorem 1.1, namely the Hölder-continuity of the function $\alpha\mapsto h(T_{\alpha})$.

The first step is to prove an estimate of the essential spectral radius of the transfer operator acting on the space of BV functions (Lasota-Yorke inequality). If one can prove a bound which is uniform in $\alpha$, then the invariant densities $\rho_{\alpha}$ turn out to be continuous in the $L^{1}$-topology and their $BV$-norms are bounded. This method has been undertaken in [10], but unfortunately their estimates prove to be too optimistic111The mistake in [10] consists in using, in eq. (12), the estimate (1) of lemma 6.1 of the present paper on the sets $\tilde{I}_{\xi}^{(n)}$, which are not intervals if $n>1$.: the bulk of section 3.1 (prop. 3.1) is another proof of this uniform Lasota-Yorke inequality.

The second step (section 3.2) is to estimate the modulus of continuity of $h(T_{\alpha})$: we will prove Hölder-continuous dependence of the invariant densities $\rho_{\alpha}$ in the $L^{1}$-topology, by using a stability result for the spectral projectors [7]. The theorem then follows from Rohlin’s formula.

3.1. Spectral radius estimate

We are going to give a proof of the following uniform Lasota-Yorke inequality (in order to simplify notation, from now on $\operatorname*{Var}_{I_{\alpha}}f$ will just be denoted $\operatorname*{Var}f$):

Proposition 3.1**.**

Let $\underline{\alpha}\in(0,1)$. Then there exist a neighbourhood $U$ of $\underline{\alpha}$ and constants $0<\lambda<1,C>0,D>0$ such that for every $\alpha\in U$

[TABLE]

Although several inequalities of this type are present in the literature, (i.e. in [15]), these are generally given for individual maps. However, for the goal of this paper it is absolutely essential that coefficients $\lambda,C,D$ can be chosen uniformly in $\alpha$, hence one needs to take this dependence into account. As $\alpha$ moves, even just in a neighbourhood of some fixed $\underline{\alpha}$, topological bifurcations are present (for instance if $\underline{\alpha}$ is a fixed point of some branch of $T_{\underline{\alpha}}$) hence in the formula (3) new boundary terms appear, requiring a very careful control.

Lemma 3.2**.**

For each $\alpha\in(0,1)$, for each $f\in BV(I_{\alpha})$

[TABLE]

Proof.

[TABLE]

[TABLE]

∎

Observe that $g_{n,\alpha}$ has infinitely many jumps discontinuities (indeed it is zero on the boundary of any interval of the partition $\mathcal{P}_{n}$), but all those jumps sum up to a finite total variation. We will, however, need to prove the stronger statement that the total variation of $g_{n,\alpha}$ decays exponentially fast in $n$, and uniformly in $\alpha$. The idea of the proof is to control the total variation of $g_{n,\alpha}$ by writing it as a sum of two functions, $h_{n,\alpha}$ and $l_{n,\alpha}$