THERMODYNAMIC FORMALISM BEYOND UNIFORMLY HYPERBOLIC SYSTEMS
Open Access
- Author:
- Zelerowicz, Agnieszka
- Graduate Program:
- Mathematics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- July 08, 2019
- Committee Members:
- Yakov B Pesin, Dissertation Advisor/Co-Advisor
Yakov B Pesin, Committee Chair/Co-Chair
Svetlana Katok, Committee Member
Misha Guysinsky, Committee Member
Igor S Aronson, Outside Member - Keywords:
- thermodynamic formalism
equilibrium measures
nonuniform hyperbolicity
partial hyperbolicity - Abstract:
- The aim of this thesis is to study thermodynamics of differentiable dynamical systems which display various forms of hyperbolicity. In the classical, uniformly hyperbolic setting we propose a new way of constructing equilibrium measures which uses some tools from geometric measure theory. We then use it to construct equilibrium measures for a certain class of partially hyperbolic systems. Almost immediately from the construction one obtains that those measures are unique equilibrium measures with full support which have local product structure. We also consider systems which are nonuniformly hyperbolic We present a systematic study of thermodynamics formalism by methods of inducing. We give a complete overview of known results and prove additional ones. One of the main contributions is the proof of Bernoulli property for equilibrium measures corresponding to a certain class of potential functions. We also prove analyticity of the pressure function for a certain class of families of potential functions. Finally,we study a certain class of nonuniformly hyperbolic attractors. Those are attractors obtained from uniformly hyperbolic systems by the slow down procedure. We study the family of geometric t-potentials on some interval in t. We show existence and uniqueness of equilibrium measures for potentials in this family. We prove that those equilibrium measures satisfy the Central Limit Theorem, have exponential decay of correlations, and the Bernoulli property. Finally, we prove that the pressure function is real analytic.