Equilibrium Measures for Symbolic Spaces Using Dimension Theory

The theory of thermodynamic formalism is well developed for subshifts of finite type. However, less is known about equilibrium measures for more general shift spaces beyond existence and, in some cases, uniqueness. Rather than constructing equilibrium measures as a limit of a sequence of measures or via a fixed point theorem, equilibrium measures can be constructed using dimension theoretic techniques. This approach can be thought of as a generalization of Hausdorff measure, and we outline a number of criteria that guarantee that this construction produces an equilibrium state. For two-sided shifts, we will show that subshifts of finite type satisfy these conditions. We will also show that equilibrium measures can be constructed for shifts with synchronizing words under additional assumptions. Another notable application of this approach is a construction of the measure of maximal entropy for finite horizon dispersing billiards. For one-sided subshifts of finite type, we construct the eigenmeasure and eigenfunction that appear in the Ruelle-Peron-Frobenius theorem using this dimension theoretic machinery.

Name thesis:degree_name

Doctor of Philosophy

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Houston

Year dc:date.issued

2025

Author dc:creator

• Day, Jason 1995-

Advisor dc:contributor.advisor

• Climenhaga, Vaughn

Committee members dc:contributor.committeemember

• Torok, Andrew
• Nicol, Matthew
• Gunaratne , Gemunu

• Mathematics

Language dc:language.iso

English

Handle dc:identifier.uri

https://hdl.handle.net/10657/19574

OAI identifier oai:identifier

oai:uh-ir.tdl.org:10657/19574