Dynamical properties of the negative beta transformation

TL;DR

This paper investigates the dynamical properties of the negative beta transformation, proving its exactness for all beta>1, and explores its connections to number theory and sequence behavior, including the Thue-Morse sequence.

Contribution

It proves the exactness of the negative beta transformation for all beta>1, confirming a conjecture, and links the limit behavior of expansions to the Thue-Morse sequence, extending properties of Parry numbers.

Findings

The negative beta transformation is exact for all beta>1.

The limit behavior of the expansion of 1 relates to the Thue-Morse sequence.

Every Yrrap number is a Perron number, extending properties of Parry numbers.

Abstract

We analyse dynamical properties of the negative beta transformation, which has been studied recently by Ito and Sadahiro. Contrary to the classical beta transformation, the density of the absolutely continuous invariant measure of the negative beta transformation may be zero on certain intervals. By investigating this property in detail, we prove that the $(-\beta)$-transformation is exact for all $\beta>1$, confirming a conjecture of G\'ora, and intrinsic, which completes a study of Faller. We also show that the limit behaviour of the $(-\beta)$-expansion of 1 when $\beta$ tends to 1 is related to the Thue-Morse sequence. A consequence of the exactness is that every Yrrap number, which is a $\beta>1$ such that the $(-\beta)$-expansion of 1 is eventually periodic, is a Perron number. This extends a well-known property of Parry numbers. However, the set of Parry numbers is different from the set of Yrrap numbers.