Level-2 large deviation principle for countable Markov shifts without Gibbs states

TL;DR

This paper establishes a level-2 large deviation principle for countable Markov shifts without relying on Gibbs states, using inducing techniques to handle non-compact sets, with applications to continued fractions and quadratic irrationals.

Contribution

It introduces a new approach to prove large deviation principles for countable Markov shifts without Gibbs states by employing inducing methods for exponential tightness.

Findings

Established level-2 large deviation principle for countable Markov shifts

Applied results to continued fraction expansions and Rénye map

Proved weighted equidistribution of quadratic irrationals

Abstract

We consider level-2 large deviations for the one-sided countable full shift without assuming the existence of Bowen's Gibbs state. To deal with non-compact closed sets, we provide a sufficient condition in terms of inducing which ensures the exponential tightness of a sequence of Borel probability measures constructed from periodic configurations. Under this condition we establish the level-2 Large Deviation Principle. We apply our results to the continued fraction expansion of real numbers in $[0,1)$ generated by the R\'enyi map, and obtain the level-2 Large Deviation Principle, as well as a weighted equidistribution of a set of quadratic irrationals to equilibrium states of the R\'enyi map.