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 Hölder 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 $.