2019/07/31 by Artur O. Lopes, Victor Vargas
Computer Science · Mathematics · Physics and Astronomy · #Alphabet #BETA (programming language) #Base (topology) #Cellular Automata and Applications #Combinatorics #Compact space #Eigenfunction #Eigenvalues and eigenvectors #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Operator (biology) #Physics #Quantum chaos and dynamical systems #Quantum mechanics #Topological entropy #math.DS #math.PR #msc:11A63 #msc:28Dxx #msc:37A35 #msc:37D35
paper · pdf · doi:10.5565/publmat6422012
published as Publ. Mat. 64 (2), pp. 661-680, 2020
arxiv created 2019/10/23 · openalex publication_date 2020/06/29 · arxiv updated 2021/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Consider m \∈ \ℕ and \β \∈ (1, m + 1]. Assume that a\∈ \ℝ can be represented in base \β using a development in series a = \∑\∞n = 1x(n)\β-n, where the sequence x = (x(n))n \∈ \ℕ takes values in the alphabet \Am := 0, dotsc, 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 \∈ \ℕ \∈ \Am^\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 Xm,\β \⊂\Am^\ℕ. The space Xm, \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 colon Xm, \β \→\ℝ, we consider the Ruelle operator \LA and we show the existence of a positive eigenfunction \ψA and an eigenmeasure \ρA for some values of m and \β. We also consider a variational principle of pressure. Moreover, we prove that the family of entropies (h(\μtA))t>0 converges, when t \→\∞, to the maximal value among the set of all possible values of entropy of all A-maximizing probabilities.