On involution kernels and large deviations principles on $\beta$-shifts

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, denoted by $\Sigma_\beta$, 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 to the bilateral $\beta$-shift of its corresponding Gibbs state $\mu_A$. Our main goal here is to prove a first level large deviations principle for the family $(\mu_{tA})_{t>1}$ with a rate function $I$ 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 $\Sigma_\beta$ and its bilateral extension $\widehat{\Sigma_\beta}$ in terms of the quasi-greedy $\beta$-expansion of $1$ and the so called involution kernel associated to the potential $A$.