Countable Alphabet Random Subhifts of Finite Type with Weakly Positive Transfer Operator

Abstract

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 [5]), 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.

Equilibrium states for non-transitive random open and closed dynamical systems

“…For instance, Buzzi [6,§0.2] noted difficulties in decomposing one-dimensional piecewise expanding random systems into pathwise irreducible components, and hence in the search for decompositions that could play the role of normal forms in this setting. Accordingly, the study of decay of correlations and Perron-Frobenius type results in the random setting has so far relied on stronger hypotheses, such as mixing and/or covering conditions [5,4,6,15,20,19,10,3,14,1,22,2]. Similar assumptions appear in the investigation of memory loss in time-dependent systems [21,23,13,7].…”

Thermodynamic Formalism for Random Weighted Covering Systems

“…For random subshifts of finite type, Bogenschütz and Gundlach [7] and Gundlach [21] proved uniqueness of (random) relative equilibrium states. Thermodynamic formalism for countable random Markov shifts has been investigated in various settings by Denker, Kifer, and Stadlbauer [15], Stadlbauer [42,43], Roy and Urbański [38], and Mayer and Urbański [32]. Mayer, Skorulski, and Urbański developed distance expanding random mappings in [33], which generalize the works of [25] and [7].…”

Thermodynamic formalism for random non-uniformly expanding maps

“…Bogenschütz and Gundlach [10] established a Ruelle theorem for random subshifts of finite type. There are also contributions to the thermodynamic formalism of countable Markov shifts, by Denker, Kifer and Stadlbauer [18], Stadlbauer [48,49], Mayer and Urbański [38], of random countable iterated function systems with overlaps, by Mihailescu and Urbański [40], and random dynamics of hyperbolic entire and meromorphic functions of finite order satisfying a growth condition at infinity, by Mayer and Urbański [39], among many others. For the construction of annealed equilibrium states and description of their decay rate in the context of random expanding maps we refer the reader to [6,50].…”