Maximizing measures in hyperbolic dynamics

In the study of dynamical systems, the long term asymptotic behaviour is often best understood in terms of properties of invariant measures. In the particular setting of Lagrangian flows, Mane and Mather formulated a number of important questions about those measures whose integrals maximized a particular integral. As part of a general theory, one can ask similar questions about quite diverse dynamical systems. In the particular case of hyperbolic (or chaotic) dynamical systems, we can consider a natural class of measures called Gibbs measures, whose study originated in the mathematical theory of Statistical Mechanics, and the work of Ruelle and Sinai. This proposal relates to how maximizing measures for typical functions for these hyperbolic systems can be approximated by these better behaved Gibbs measures. This has important implications for understanding the quite complicated, but important, maximizing measures in terms of much simpler Gibbs measures.