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