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 $\alpha \in \Bigl[0, \frac{\lfloor \beta \rfloor}{\beta - 1}\Bigr]$canberepresentedinbase$\beta$ using a development in series of the form $\alpha ={\sum }_{n=1}^{\mathrm{\infty }}{x}_n{\beta }^{-n}$, where $x=(x_n{)}_{n\ge 1}$ is a sequence taking values into the alphabet $\{0,\mathrm{.}\mathrm{.}\mathrm{.},\lfloor \beta \rfloor \}$. The so called $\beta$-shift, denoted by ${\mathrm{\Sigma }}_{\beta }$, 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 ${\mathrm{\Sigma }}_{\beta }$ and its bilateral extension $\widehat{{\mathrm{\Sigma }}_{\beta }}$ in terms of the quasi-greedy $\beta$-expansion of $1$ and the so called involution kernel associated to the potential $A$.