  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.
