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

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 Teichmuller theory for Riemann surfaces. Furthermore, we prove that the space of those continuous potentials has a Teichmuller structure. Moreover, this Teichmuller structure is a complete structure and is the completion of the space of smooth potentials under this Teichmuller 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.