Teichmüller Structures and Dual Geometric Gibbs Type Measure Theory for Continuous Potentials

Abstract

The Gibbs measure theory for smooth potentials is an old and beautiful subject and has many important applications in modern dynamical systems. For continuous potentials, it is impossible to have such a theory in general. However, we develop a dual geometric Gibbs type measure theory for certain continuous potentials in this paper following some ideas and techniques from Teichmüller theory for Riemann surfaces. Furthermore, we prove that the space of those continuous potentials has a Teichmüller structure. Moreover, this Teichmüller structure is a complete structure and is the completion of the space of smooth potentials under this Teichmüller structure. Thus our dual geometric Gibbs type theory is the completion of the Gibbs measure theory for smooth potentials from the dual geometric point of view.

References in corpus (3)

• Differentiating the absolutely continuous invariant measure of an interval map f with respect to f
• Analyticity of the Susceptibility Function for Unimodal Markovian Maps of the Interval
• Asymptotically Affine and Asymptotically Conformal Circle Endomorphisms

Cited by in corpus (1)

• Symmetric Rigidity for Circle Endomorphisms with Bounded Geometry