  Consider $\beta > 1$ and $\lfloor \beta \rfloor$ its integer part. It is
widely known that any real number $\alpha \in \Bigl[0, \frac{\lfloor \beta
\rfloor}{\beta - 1}\Bigr]$ can be represented in base $\beta$ using a
development in series of the form $\alpha = \sum_{n = 1}^\infty x_n\beta^{-n}$,
where $x = (x_n)_{n \geq 1}$ is a sequence taking values into the alphabet
$\{0,\; ...\; ,\; \lfloor \beta \rfloor\}$. The so called $\beta$-shift
$\Sigma_\beta$ is given as the set of sequences such that all their iterates by
the shift map are less than or equal to the quasi-greedy $\beta$-expansion of
$1$. Fixing a H\"older continuous potential $A$, we show an explicit expression
for the main eigenfunction of the Ruelle operator $\psi_A$ in order to obtain a
natural extension of its corresponding Gibbs state $\mu_A$ to the bilateral
$\beta$-shift. Our main goal here is to prove a first level large deviations
principle for the family $(\mu_{tA})_{t>1}$ with a rate function attaining its
maximum value on the union of the supports of all the maximizing measures of
$A$. The above is proved through a technique using the representation of the
$\beta$-shift $\Sigma_\beta$ and its bilateral extension
$\widehat{\Sigma_\beta}$ in terms of the greedy $\beta$-expansion of $1$ and
the so called involution kernel associated to the potential $A$.
