Galois conjugates for some family of generalized beta-maps

TL;DR

This paper investigates the Galois conjugates of Yrrap numbers, a class of numbers related to negative beta-transformations, and explores their properties and analogies with positive beta-transformations.

Contribution

It determines the closure of the Galois conjugates of Yrrap numbers and establishes analogies with piecewise linear maps obtained by altering beta-transformations.

Findings

The set of Galois conjugates of Yrrap numbers is characterized.

Both non-Parry Yrrap numbers and non-Yrrap Parry numbers are countable.

Analogies are drawn between Yrrap numbers and certain piecewise linear maps.

Abstract

A real number $\beta>1$ is called an Yrrap (or Ito-Sadahiro) number if the corresponding negative $\beta$-transformation defined by $x\mapsto 1-\{\beta x\}$ for $x\in[0,1]$, where $\{y\}$ denotes the fraction part of $y\in\mathbb{R}$, has a finite orbit at $1$. Yrrap numbers are an analogy of Parry numbers for positive $\beta$-transformations given by $x\mapsto \{\beta x\}$ for $x\in[0,1]$, $\beta>1$. In this paper, we determine the closure of the set of Galois conjugates of Yrrap numbers. In addition, we show an analogy of the result to the family of piecewise linear continuous maps each of which is obtained by changing the odd-numbered branches (left-most one is regarded as $0$-th) of the $\beta$-transformation to negative ones for $\beta>1$. As an application, we see that both the set of Yrrap numbers which are non-Parry numbers and that of Parry numbers which are non-Yrrap numbers are countable.