Entropy, topological transitivity, and dimensional properties of unique 𝑞-expansions

Abstract

LetMMbe a positive integer andq∈(1,M+1].q \in (1,M+1].We consider expansions of real numbers in baseqqover the alphabet{0,…,M}\{0,\ldots , M\}. In particular, we study the setUq\mathcal {U}_{q}of real numbers with a uniqueqq-expansion, and the setUq\mathbf {U}_qof corresponding sequences.It was shown by Komornik, Kong, and Li that the functionHH, which associates to eachq∈(1,M+1]q\in (1, M+1]the topological entropy ofUq\mathcal {U}_q, is a Devil’s staircase. In this paper we explicitly determine the plateaus ofHH, and characterize the bifurcation setE\mathscr {E}ofqq’s where the functionHHis not locally constant. Moreover, we show thatE\mathscr {E}is a Cantor set of full Hausdorff dimension. We also investigate the topological transitivity of a naturally occurring subshift(Vq,σ),(\mathbf {V}_q, \sigma ),which has a close connection with open dynamical systems. Finally, we prove that the Hausdorff dimension and box dimension ofUq\mathcal {U}_qcoincide for allq∈(1,M+1]q\in (1,M+1].