Combinatorial properties of lazy expansions in Cantor real bases

TL;DR

This paper explores the combinatorial and dynamical properties of lazy expansions in Cantor bases, generalizing classical algorithms, and characterizes when these expansions form sofic shifts, with specific results for periodic Cantor bases.

Contribution

It introduces a generalized lazy expansion algorithm for Cantor bases, provides a Parry-like criterion, and characterizes soficity in periodic Cantor bases, extending previous work on greedy expansions.

Findings

Lazy expansions are obtained by flipping greedy expansion digits.

A Parry-like criterion characterizes lazy expansion sequences.

Lazy shift is sofic iff all quasi-lazy expansions are ultimately periodic.

Abstract

The lazy algorithm for a real base $\beta$ is generalized to the setting of Cantor bases $\boldsymbol{\beta}=(\beta_n)_{n\in \mathbb{N}}$ introduced recently by Charlier and the author. To do so, let $x_{\boldsymbol{\beta}}$ be the greatest real number that has a $\boldsymbol{\beta}$-representation $a_0a_1a_2\cdots$ such that each letter $a_n$ belongs to $\{0,\ldots,\lceil \beta_n \rceil -1\}$. This paper is concerned with the combinatorial properties of the lazy $\boldsymbol{\beta}$-expansions, which are defined when $x_{\boldsymbol{\beta}}<+\infty$. As an illustration, Cantor bases following the Thue-Morse sequence are studied and a formula giving their corresponding value of $x_{\boldsymbol{\beta}}$ is proved. First, it is shown that the lazy $\boldsymbol{\beta}$-expansions are obtained by "flipping" the digits of the greedy $\boldsymbol{\beta}$-expansions. Next, a Parry-like criterion characterizing the sequences of non-negative integers that are the lazy $\boldsymbol{\beta}$-expansions of some real number in $(x_{\boldsymbol{\beta}}-1,x_{\boldsymbol{\beta}}]$ is proved. Moreover, the lazy $\boldsymbol{\beta}$-shift is studied and in the particular case of alternate bases, that is the periodic Cantor bases, an analogue of Bertrand-Mathis' theorem in the lazy framework is proved: the lazy $\boldsymbol{\beta}$-shift is sofic if and only if all quasi-lazy $\boldsymbol{\beta}^{(i)}$-expansions of $x_{\boldsymbol{\beta}^{(i)}}-1$ are ultimately periodic, where $\boldsymbol{\beta}^{(i)}$ is the $i$-th shift of the alternate base $\boldsymbol{\beta}$.