Equilibrium States for the Random β-Transformation through g-Measures

2021/04/23 by Karma Dajani, Dajani, Karma, Kieran Power +1
Mathematics · #28D05 #37A05 #37A45 #37E05 #37E15 #Dynamical Systems (math.DS) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Approximation and Integration #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2104.11634

openalex publication_date 2021/04/23 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

We consider the random β-transformation K_β, defined on \0,1\^{\mathbb N}×[0, (\lfloorβ\rfloor)/(β-1)], that generates all possible expansions of the form x=∑_i=0^∞(a_i)/(βⁱ), where a_i∈ \0,1,⋯,\lfloorβ\rfloor\. This transformation was first introduced by Dajani and Kraaikamp, and later studied by Dajani and de Vries, where two natural invariant ergodic measures were found. The first is the unique measure of maximal entropy, and the second is a measure of the form m_p× μ_β, with m_p the Bernoulli (p,1-p) product measure and μ_β is a measure equivalent to Lebesgue measure. In this paper, we give an uncountable family of K_β-invariant exact g-measures for a certain collection of algebraic β's.

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