Thermodynamic Formalism for Random Weighted Covering Systems

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

“…Thus, to find a (random) invariant measure, one should normalize µ ω := qωνω νω(qω) . This work complements previous works of the authors [1,2], where they have developed a general thermodynamic formalism for random open and closed dynamical systems, without strongly contracting assumption of this work, but imposing covering type conditions. The present approach also incorporates the use of a random family of cones, a strategy previously used in [15], references therein, and recently in [22].…”

“…The proof of the next result follows similarly to the proof of Theorem 2.23 in [1] (see also Remark 2.24, Lemma 12.2 and Lemma 12.3). Theorem 6.4.…”

“…ω ) − log b ω,f dm < 0; or, slightly more generally, log g ω ∞ dm − ϕ − + log(2 + ξ (1) ω )dm − β f < 0.…”

“…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].…”

“…This work may also be regarded as a generalisation, complementary to [2], of the work of Liverani and Maume-Deschamps [16] to the random setting. Furthermore, the novelty of our approach allows us prove results for both open and closed settings at once in a concise manner in contrast to techniques used in [1,2].…”