Large deviation principle for the backward continued fraction expansion (Q2066962)

By the result in [\textit{M. Denker} and \textit{Z. Kabluchko}, Probab. Math. Stat. 27, No. 1, 139--149 (2007; Zbl 1125.60023)], for any non-constant bounded function \(\psi:\mathbb Z_{\ge 1}\to\mathbb R\), the large deviation principle holds for the mean process for the regular continued fraction expansion. The purpose of this paper is to establish the large deviation principle for the backward continued fraction expansion. One of the main results is stated as follows. For a non-constant function \(\psi:\mathbb Z_{\ge 2}\to\mathbb R\) the large deviation principle holds. The rate function is a convex function. Another result gives a complete description of the set of minimizers for any non-negative function \(\psi:\mathbb Z_{\ge 2}\to\mathbb R\), showing a difference in the regular continued fraction and backward continued fraction expansions from the viewpoint of large deviations. As a corollary to these results, the result for the three specific means for the backward continued fraction expansion is obtained. The proof of the main results rely on the analysis of the dynamics of the Rényi map. The method of proof of the first main result uses the ergodic theory and the thermodynamic formalism for topological Markov shifts as its main tool. The method gives a representation of the rate function in terms of entropy and Lyapunov exponent of invariant probability measures. For the proof of the second main result, this representation is used together with results on the multifractal analysis for the Rényi map.