arXiv 1907.046562019-07math.DSmath.PR

对称$β$移位的Ruelle算子

The Ruelle operator for symmetric $β$-shifts

Artur O. Lopes, Victor Vargas

发表信息：: Publ. Mat. 64 (2), pp. 661-680, 2020
DOI：: 10.5565/PUBLMAT6422012

摘要：

考虑$m \in \mathbb{N}$和$β\in (1, m + 1]$。假设$a\in \mathbb{R}$可以使用系列$a = \sum^{\infty}_{n = 1}x(n)β^{-n}$中的一个发展在基本$β$中表示，其中序列$x = (x(n))_{n \in \mathbb{N}}$采用字母表$\mathcal{A}_m := \{0, \ldots, m\}$中的值。上面的表达式被称为$a$的$β$ -扩展，它不一定是唯一的。我们感兴趣的是序列$x = (x(n))_{n \in \mathbb{N}} \in \mathcal{A}_m^\mathbb{N}$，它与所有可能的值$a$相关联，这些值具有唯一的扩展。我们用$X_{m,β} \subset\mathcal{A}_m^\mathbb{N}$来表示这样的$x$的集合（具有一些更多的技术限制）。空间$X_{m, β}$被称为与$(m, β)$对相关联的对称$β$移位。它是移位映射不变的，但一般来说它不是有限型的子移位。给定一个Hölder连续势$A:X_{m, β} \to\mathbb{R}$，考虑Ruelle算子$\mathcal{L}_A$，证明了当$m$和$β$取适当值时，存在正的本征函数$ψ_A$和本征测度$ρ_A$.我们还考虑了压力的变分原理。此外，我们证明了家庭的熵$h(μ_{tA})_{t>1}$收敛，当$t \to\infty$，所有可能的值的熵的所有$A$最大化概率的集合中的最大值。

英文摘要：

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

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