Narrow Results

We return to the thermodynamic formalism constructions for random expanding in average transformations and for random subshifts of finite type with random rates of topological mixing, as well as to the Perron–Frobenius type theorem for certain random positive linear operators. Our previous expositions in [14, 19] and [21] were based on constructions which left some gaps and inaccuracies related to the measurability and uniqueness issues. Our approach here is based on Hilbert projective norms which were already applied in [5] for the thermodynamic formalism constructions for random subshifts of finite type but our method is somewhat different and more general so that it enables us to treat simultaneously both expanding and subshift cases.

Let Δ ⊊ V be a proper subset of the vertices V of the defining graph of an irreducible and aperiodic shift of finite type . Let Σ_Δ be the subshift of allowable paths in the graph of which only passes through the vertices of Δ. For a random point x chosen with respect to an equilibrium state μ of a Hölder potential φ on , let τ_n be the point process defined as the sum of Dirac point masses at the times k > 0, suitably rescaled, for which the first n-symbols of T^kx belong to Δ. We prove that this point process converges in law to a marked Poisson point process of constant parameter measure. The scale is related to the pressure of the restriction of φ to Σ_Δ and the parameters of the limit law are explicitly computed.

We consider $\psi$-mixing dynamical systems $({𝒳},T,{{ℬ}}_{{𝒳}},\mu )$ and we find conditions on families of sets $\{{{𝒰}}_n\subset {𝒳}:n\in \mathbb{N}\}$ so that $\mu ({{𝒰}}_n){\tau }_n$ tends in law to an exponential random variable, where ${\tau }_n$ is the entry time to ${{𝒰}}_n.$

We study the thermodynamic formalism for particular types of sub-additive sequences on a class of subshifts over countable alphabets. The subshifts we consider include factors of irreducible countable Markov shifts under certain conditions, which we call irreducible countable sofic shifts. We show the variational principle for topological pressure for some sub-additive sequences with tempered variation on irreducible countable sofic shifts. We also study conditions for the existence and uniqueness of invariant ergodic Gibbs measures and the uniqueness of equilibrium states. Applications are given to some dimension problems and study of factors of (generalized) Gibbs measures on certain subshifts over countable alphabets.