An Introduction to Geometric Gibbs Theory

This is an article I wrote for Dynamics, Games, and Science. In Dynamics, Game, and Science, one of the most important equilibrium states is a Gibbs state. The deformation of a Gibbs state becomes an important subject in these areas. An appropriate metric on the space of underlying dynamical systems is going to be very helpful in the study of deformation. The Teichmüller metric becomes a natural choice. The Teichmüller metric, just like the hyperbolic metric on the open unit disk, makes the space of underlying dynamical systems a complete space. The Teichmüller metric precisely measures the change of the eigenvalues at all periodic points which are essential data needed to obtain the Gibbs state for a given dynamical system. In this article, I will introduce the Teichmüller metric and, subsequently, a generalization of Gibbs theory which we call geometric Gibbs theory.