Galois conjugates for some family of generalized beta-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.