The Ruelle operator for symmetric $\beta$-shifts

Abstract

Consider $m \in \mathbb{N}$ and $\beta \in (1, m + 1]$. Assume that $a\in \mathbb{R}$ can be represented in base $\beta$ using a development in series $a = \sum^{\infty}_{n = 1}x(n)\beta^{-n}$, where the sequence $x = (x(n))_{n \in \mathbb{N}}$ takes values in the alphabet $\mathcal{A}_m := \{0, \dotsc, m\}$. The above expression is called the $\beta$-expansion of $a$ and it is not necessarily unique. We are interested in sequences $x = (x(n))_{n \in \mathbb{N}} \in \mathcal{A}_m^\mathbb{N}$ which are associated to all possible values $a$ which have a unique expansion. We denote the set of such $x$ (with some more technical restrictions) by $X_{m,\beta} \subset\mathcal{A}_m^\mathbb{N}$. The space $X_{m, \beta}$ is called the symmetric $\beta$-shift associated to the pair $(m, \beta)$. It is invariant by the shift map but in general it is not a subshift of finite type. Given a Hölder continuous potential $A\colon X_{m, \beta} \to\mathbb{R}$, we consider the Ruelle operator $\mathcal{L}_A$ and we show the existence of a positive eigenfunction $\psi_A$ and an eigenmeasure $\rho_A$ for some values of $m$ and $\beta$. We also consider a variational principle of pressure. Moreover, we prove that the family of entropies $(h(\mu_{tA}))_{t>0}$ converges, when $t \to\infty$, to the maximal value among the set of all possible values of entropy of all $A$-maximizing probabilities.