Math 7394

Ergodic theory and thermodynamic formalism
Spring 2019

• Office: 665 PGH
• Office hours:  Monday and Wednesday 2-2:50pm
  
• Email: climenha [at] math.uh.edu

• Lectures:  MWF, 11am-12pm, AH 2
  
• Textbook:  An Introduction to Ergodic Theory, by Peter Walters
  
• Supplementary references:
  
  • Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, by Rufus Bowen
  • Other research and survey papers and supplementary notes to be posted and distributed
    
• Syllabus
  

Ergodic theory is a central part of the theory of dynamical systems, studying the asymptotic statistical properties of systems evolving in time that preserve an invariant measure.  Systems with chaotic behavior generally possess many invariant measures, and thermodynamic formalism borrows tools from statistical mechanics to select a distinguished measure that is physically relevant.  The first part of the class will cover topics in classical ergodic theory, including Birkhoff's ergodic theorem, entropy, and the classification of Bernoulli automorphisms. 

The remainder of the course will discuss thermodynamic formalism, including the description of Sinai-Ruelle-Bowen measure via absolute continuity, the description of Parry measure via a variational principle, and the connection between the two via the general theory of equilibrium states.  Some time will be spent describing the different approaches to thermodynamic formalism and SRB measures in uniform hyperbolicity: Ruelle-Perron-Frobenius operators indirectly via symbolic dynamics or directly via anisotropic Banach spaces; specification and expansivity; and the geometric approach via averaged pushforwards.  Time permitting, we will discuss connections to dimension theory and geometric measure theory, and will conclude with a discussion of the nonuniformly hyperbolic setting.