d-bar continuity beyond subshifts of finite type

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 $\overline{d}$-metric on the space of measures and the metric coming from the Hölder norm on the space of potentials. The concept of $\overline{d}$-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 $\overline{d}$-metric. Many ergodic properties are well-behaved with respect to this metric. The entropy function μ \mapsto h_μ(σ) is $\overline{d}$-continuous. Moreover, the set of processes that are isomorphic to Bernoulli shifts is $\overline{d}$-closed. The more familiar topology on the space of Borel probability measures is the weak*-topology. The topology coming from the $\overline{d}$-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 $\phi$ to its unique equilibrium state μ_\phi is continuous with respect to the $\overline{d}$-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.

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Houston

Year dc:date.issued

2023

Author dc:creator

• Bhullar, Jasmine

Advisor dc:contributor.advisor

• Climenhaga, Vaughn

Committee members dc:contributor.committeemember

• Nicol, Matthew
• Ott, William
• Gunaratne, Gemunu H.

• Dynamical Systems
• Thermodynamic Formalism
• Ergodic Theory

Statement dc:rights

• The author of this work is the copyright owner. UH Libraries and the Texas Digital Library have their permission to store and provide access to this work. Further transmission, reproduction, or presentation of this work is prohibited except with permission of the author(s).

Language dc:language.iso

eng

Handle dc:identifier.uri

https://hdl.handle.net/10657/15930

OAI identifier oai:identifier

oai:uh-ir.tdl.org:10657/15930