  Consider $m \in \mathbb{N}$ and $\beta \in (1, m + 1]$. Assume that $a\in
\mathbb{R}$ can be represented in base $\beta$ using a development in series $a
= \sum^{\infty}_{n = 1}x(n)\beta^{-n}$ where the sequence $x = (x(n))_{n \in
\mathbb{N}}$ take values in the alphabet $\mathcal{A}_m := \{0, \ldots, m\}$.
The above expression is called the $\beta$-expansion of $a$ and it is not
necessarily unique. We are interested in sequences $x = (x(n))_{n \in
\mathbb{N}} \in \mathcal{A}_m^\mathbb{N}$ which are associated to all possible
values $a$ which have a unique expansion. We denote the set of such $x$ (with
some more technical restrictions) by $X_{m,\beta}
\subset\mathcal{A}_m^\mathbb{N}$. The space $X_{m, \beta}$ is called the
symmetric $\beta$-shift associated to the pair $(m, \beta)$. It is invariant by
the shift map but in general it is not a subshift of finite type. Given a
H\"older continuous potential $A:X_{m, \beta} \to\mathbb{R}$, we consider the
Ruelle operator $\mathcal{L}_A$ and we show the existence of a positive
eigenfunction $\psi_A$ and an eigenmeasure $\rho_A$ for some appropriated
values of $m$ and $\beta$. We also consider a variational principle of
pressure. Moreover, we prove that the family of entropies $h(\mu_{tA})_{t>1}$
converges, when $t \to\infty$, to the maximal value among the set of all
possible values of entropy of all $A$-maximizing probabilities.
