d-bar continuity beyond subshifts of finite type

Date
2023-08
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract

The main result of this thesis is that for a mixing countable state Markov shift, the map taking a strongly positive recurrent potential to its Ruelle-Perron-Frobenius (RPF) measure is continuous in the d―-metric on the space of measures and the metric coming from the Hölder norm on the space of potentials. The concept of d―-distance on the space of invariant measures on a shift space was introduced by Ornstein to study the isomorphism problem for Bernoulli shifts. Ornstein showed that entropy is a complete invariant for the class of Bernoulli shifts: two Bernoulli shifts are isomorphic if and only if they have the same entropy. A key tool in this proof was the d―-metric. Many ergodic properties are well-behaved with respect to this metric. The entropy function μ↦hμ(σ) is d―-continuous. Moreover, the set of processes that are isomorphic to Bernoulli shifts is d―-closed. The more familiar topology on the space of Borel probability measures is the weak*-topology. The topology coming from the d―-metric refines the weak*-topology. For Topological Markov Shifts on finite alphabets, potentials with summable variations have a unique equilibrium state. One can quickly prove that the map that sends a potential to its unique equilibrium state is continuous in the weak*-topology. For full shifts on a finite alphabet, Coelho and Quas proved that the map that sends a potential ϕ to its unique equilibrium state μϕ is continuous with respect to the d―-metric on the space of shift-invariant probability measures and a suitable metric on the space of potentials. We extend this result to the setting of full shifts on countable (infinite) alphabets, and to mixing Countable-state Markov Shifts. As part of the proof, we show that the map that sends a strongly positive recurrent potential to its normalization is continuous for potentials on mixing Countable-state Markov Shifts.

Description
Keywords
Dynamical Systems, Thermodynamic Formalism, Ergodic Theory
Citation