Date of Award
Spring 6-13-2025
Document Type
Dissertation
Degree Name
Ph.D. in Mathematics
Organizational Unit
College of Natural Science and Mathematics, Mathematics
First Advisor
Ronnie Pavlov
Second Advisor
Kingshuk Ghosh
Third Advisor
Nicholas Ormes
Fourth Advisor
Anh Le
Copyright Statement / License for Reuse
All Rights Reserved.
Keywords
Thermodynamic formalism, Dynamical systems, dynamical systems, Computability theory, Equilibrium and freezing states
Abstract
This dissertation investigates several key questions at the intersection of dynamical systems, computability theory, and thermodynamic formalism. In the symbolic setting, we establish novel results regarding the statistical properties of equilibrium states, deriving bounds on probabilities of configurations and relating these bounds to the Gibbs property through the homoclinic relation. Additionally, we examine the computability of thermodynamic quantities such as pressure, ground state energy, and residual entropy. We show that topological pressure is computable from above for general subshifts and computable for strongly irreducible shifts, with similar results extending to ground state energy and residual entropy.
Extending beyond subshifts, we explore freezing phase transitions in general dynamical systems. We show that for a given invariant measure π, it is no more restrictive that π is the freezing state for some potential than it is for π to be the equilibrium state for some potential. Additionally we show that the set of freezing potentials is dense in the space of all potentials when the entropy map is upper semi-continuous. Together, these results contribute to the existing literature on thermodynamic formalism, deepening our understanding of equilibrium and freezing states while advancing the study of computability in this context.
Copyright Date
6-2025
Publication Statement
Copyright is held by the author. User is responsible for all copyright compliance.
Rights Holder
C. Evans Hedges
Provenance
Received from ProQuest
File Format
application/pdf
Language
English (eng)
Extent
108 pgs
File Size
471 KB
Recommended Citation
Hedges, C. Evans, "Some Results in Thermodynamic Formalism" (2025). Electronic Theses and Dissertations. 2569.
https://digitalcommons.du.edu/etd/2569