Beyond Bowen’s Specification Property

A classical result in thermodynamic formalism is that for uniformly hyperbolic systems, every Hölder continuous potential has a unique equilibrium state. One proof of this fact due to Rufus Bowen and uses the such systems satisfy expansivity specification properties. In these notes, we survey recent progress generalizations properties extend Bowen’s arguments beyond uniform hyperbolicity, including applications partially geodesic flows negative curvature. We include new criterion uniqueness states with 1-dimensional center.