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

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Authors

Colleges, School and Institutes

Abstract

Let M be a positive integer and q ∈ (1,M +1]. We consider expansions of real numbers in base q over the alphabet {0,…,M}. Inparticular, we study the set Uq of real numbers with a unique q-expansion, and the set Uq of corresponding sequences. It was shown by Komornik, Kong, and Li that the function H, which associates to each q ∈ (1,M+1] the topological entropy of Uq, is a Devil’s staircase. In this paper we explicitly determine the plateaus of H, and characterize the bifurcation set E of q’s where the function H is not locally constant. Moreover, we show that E is a Cantor set of full Hausdorff dimension. We also investigate the topological transitivity of a naturally occurring subshift (Vq,σ), which has a close connection with open dynamical systems. Finally, we prove that the Hausdorff dimension and box dimension of Uq coincide for all q ∈ (1,M +1].

Bibliographic note

?First published in Transactions of the American Mathematical Society in 371, 5. 2019 published by the American Mathematical Society,? and the copyright notice in proper form must be placed on all copies.

Details

Original languageEnglish
Pages (from-to)3209-3259
Number of pages51
JournalTransactions of the American Mathematical Society
Volume371
Issue number5
Early online date27 Nov 2018
Publication statusPublished - 1 Jan 2019

Keywords

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