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

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