Countable Alphabet Random Subhifts of finite type with weakly positive transfer operator

TL;DR

This paper develops a thermodynamical formalism for countable alphabet random subshifts of finite type without the Big Images Property, establishing existence, uniqueness, and spectral properties of invariant measures and transfer operators.

Contribution

It extends thermodynamical formalism to non-uniformly positive transfer operators in random subshifts, proving existence, uniqueness, and spectral gap results.

Findings

Existence of random conformal measures with bounds

Proven spectral gap for transfer operators

Established exponential decay of correlations

Abstract

We deal with countable alphabet locally compact random subshifts of finite type (the latter merely meaning that the symbol space is generated by an incidence matrix) under the absence of Big Images Property and under the absence of uniform positivity of the transfer operator. We first establish the existence of random conformal measures along with good bounds for the iterates of the Perron-Frobenius operator. Then, using the technique of positive cones and proving a version of Bowen's type contraction (see \cite{Bow75}), we also establish a fairly complete thermodynamical formalism. This means that we prove the existence and uniqueness of fiberwise invariant measures (giving rise to a global invariant measure) equivalent to the fiberwise conformal measures. Furthermore, we establish the existence of a spectral gap for the transfer operators, which in the random context precisely means the exponential rate of convergence of the normalized iterated transfer operator. This latter property in a relatively straightforward way entails the exponential decay of correlations and the Central Limit Theorem.