Thermodynamical Formalism For Maps With Inducing Schemes
Open Access
- Author:
- Zhang, Ke
- Graduate Program:
- Mathematics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- May 09, 2007
- Committee Members:
- Yakov B Pesin, Committee Chair/Co-Chair
Svetlana Katok, Committee Member
Omri Sarig, Committee Member
Piotr Berman, Committee Member - Keywords:
- dynamical systems
thermodynamic formalism
inducing scheme - Abstract:
- We study the thermodynamical formalism for a class of systems admitting inducing schemes. Following Pesin and Senti, we discuss the general procedure for the existence and uniqueness of equilibrium measure among the liftable class. To obtain equilibrium measure in the usual sense we need to study the liftable class, and prove that the equilibrium measure is in this class. We demonstrate two approaches to liftability: first, we use techniques of Keller and Zweimuller to study the structure of the inducing scheme directly and provide a sufficient condition and some examples. Building on a result of Buzzi, we use liftabilty result to the Markov extension of a piecewise invertible map to show the liftability of the inducing schemes that can be "embedded" to the Markov extension. We will also use this theory to study certain examples. In Chapter 2 we study uniformly expanding system with a class of non-H"older potential, prove existence and uniqueness of equilibrium measure, and an example a phase transition. In Chapter 4, we show that for systems admitting Young's tower with potentials of the type $varphi_t(x)=-tlog|det Df^u(x)|$, there is an equilibrium measure among the liftable class.