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