Thermodynamic formalism for topological Markov chains on standard Borel spaces

Abstract

We develop a Thermodynamic Formalism for bounded continuous potentials defined on the sequence space X ≡ E N , where E is a general standard Borel space. In particular, we introduce meaningful concepts of entropy and pressure for shifts acting on X and obtain the existence of equilibrium states as finitely additive probability measures for any bounded continuous potential. Furthermore, we establish convexity and other structural properties of the set of equilibrium states, prove a version of the Perron-Frobenius-Ruelle theorem under additional assumptions on the regularity of the potential and show that the Yosida-Hewitt decomposition of these equilibrium states does not have a purely finite additive part.We then apply our results to the construction of invariant measures of timehomogeneous Markov chains taking values on a general Borel standard space and obtain exponential asymptotic stability for a class of Markov operators. We also construct conformal measures for an infinite collection of interacting random paths which are associated to a potential depending on infinitely many coordinates. Under an additional differentiability hypothesis, we show how this process is related after a proper scaling limit to a certain infinite-dimensional diffusion.2010 Mathematics Subject Classification. 37D35, 28Dxx.

Entropy, pressure, ground states and calibrated sub-actions for linear dynamics

“…Following the proof of the main Theorem in [LMSV19], with a modification in the part that guarantees the existence of the main eigenfunction (we adapt the reasoning of Section 3 in [CSS19] to the linear dynamics setting), it is not difficult to check that for any bounded above potential A ∈ H α (X) ∩ SV T (X), there is an eigenvalue λ A > 0 and a strictly positive eigenfunction ψ A ∈ H b,α (X) associated with λ A . Moreover, under the assumption that L A (1) = 1, it is possible to guarantees the existence of a Gibbs state µ A (i.e.…”

Ruelle operator for continuous potentials and DLR-Gibbs measures

“…Dobrushin uniqueness theorem. In this section we prove an uniqueness theorem in the high temperature regime (β small) for potentials not satisfying Bowen's condition in (13). This result applies for a very large class of potentials which live outside the Hölder, Walters and Bowen spaces.…”

“…On the other hand, the φ 4 -model from Statistical Mechanics also motivates the study of alphabets which are Polish spaces. In this setting, equilibrium states might only exist as positive linear functionals, but for summable Hölder potentials, the Ruelle operator still has a spectral gap (see [13]).…”

Existence of Gibbs states and maximizing measures on a general one-dimensional lattice system with markovian structure

“…In [Sar99] and [MU01] these results were generalized for the non-compact setting of countable Markov shifts. Another interesting model, known as XY model, was studied in [BCL+11] and [LMST09], from which was derived some interesting generalizations for compact, bounded and even non-bounded metric spaces (see for instance [CSS19,LMMS15,LMV19,LV19]).…”

“…The uniqueness of the ground state was proved in [Kem11] in the setting of countable Markov shifts satisfying the BIP property, however, that problem is still open for the topologically transitive case. In the setting of XY models, problems of selection and non-selection at zero temperature were studied in [BCL+11] and [LMST09] for the classical approach on the interval [0, 1], and these results were generalized to compact metric spaces in [LMMS15] and to a non-compact bounded metric spaces in [CSS19] and [LV19].…”