Transformations generating negative $\beta$-expansions

TL;DR

This paper introduces a family of dynamical systems for generating negative -expansions, characterizes their digit sequences, and explores the concept of negative greedy expansions, highlighting the limitations of these systems.

Contribution

It provides a new framework for negative -expansions, characterizes the digit sequences produced, and discusses the concept of greedy expansions in the negative base context.

Findings

Support of invariant measure is absolutely continuous w.r.t. Lebesgue measure

Characterization of digit sequences produced by typical transformations

Negative greedy expansions cannot be generated by the introduced dynamical systems

Abstract

We introduce a family of dynamical systems that generate negative $\beta$-expansions and study the support of the invariant measure which is absolutely continuous with respect to Lebesgue measure. We give a characterization of the set of digit sequences that is produced by a typical member of this family of transformations. We discuss the meaning of greedy expansions in the negative sense, and show that there is no transformation in the introduced family of dynamical systems that generates negative greedy. However, if one looks at random algorithms, then it is possible to define a greedy expansion in base $-\beta$.