Thermodynamic formalism for countable symbolic Markov chains

A thermodynamic definition of topological pressure for non-compact sets

“…By corollary 1.7 of [12], P int (Σ, 0) = P ext (Σ, 0) in the metric d. The result follows from theorem 6.9. Theorem 6.13.…”

“…We note that since X is a separable space, if µ is ergodic then µ(G µ ) = 1. Thus, there exists x satisfying V(x) = µ, which shows that P * X (ϕ) ≥ c. In [12], Gurevich and Savchenko study two definitions of topological pressure adapted to non-compact spaces. We compare these with P * Z (ϕ).…”

“…Both inequalities of theorem 6.9 may be strict. As noted in [12] and [13], let Y be a compact metric space and f : Y → Y be a minimal homeomorphism with h top (f ) > 0. Let ϕ = 0.…”

“…Let P G (ϕ) denote the Gurevic pressure as defined by Sarig [22] where ϕ is a locally Hölder function and h G (σ) := P G (0). In [12], the authors allow Σ to be equipped with more general metrics and study P int (Σ, ϕ) and P ext (Σ, ϕ) for ϕ ∈ C(Σ). To rephrase corollary 1 of [22], Sarig showed that in his setting P G (ϕ) = P int (Σ, ϕ).…”

“…It has long been thought desirable to generalise the standard theory of topological pressure and equilibrium states to non-compact spaces [5], [9], [12], [13], [14], [22]. The purpose of this paper is to contribute a new and elementary approach to this problem.…”