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On involution kernels and large deviations principles on $β$-shifts

Consider $β> 1$ and $\lfloor β\rfloor$ its integer part. It is widely known that any real number $α\in \Bigl[0, \frac{\lfloor β\rfloor}{β- 1}\Bigr]$ can be represented in base $β$ using a development in series of the form $α= \sum_{n = 1}^\infty x_nβ^{-n}$, where $x = (x_n)_{n \geq 1}$ is a sequence taking values into the alphabet $\{0,\; ...\; ,\; \lfloor β\rfloor\}$. The so called $β$-shift, denoted by $Σ_β$, 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 $β$-expansion of $1$. Fixing a Hölder continuous potential $A$, we show an explicit expression for the main eigenfunction of the Ruelle operator $ψ_A$, in order to obtain a natural extension to the bilateral $β$-shift of its corresponding Gibbs state $μ_A$. Our main goal here is to prove a first level large deviations principle for the family $(μ_{tA})_{t>1}$ 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 $Σ_β$ and its bilateral extension $\widehat{Σ_β}$ in terms of the quasi-greedy $β$-expansion of $1$ and the so called involution kernel associated to the potential $A$.