Appears in collection : New directions in thermodynamic formalism / Nouvelles orientations pour le formalisme thermodynamique
Thermodynamic formalism involves many quantities that grow or decay exponentially fast. Comparing quantities with the same exponential rate results in ratios that are permitted to grow or decay sub-exponentially. A recurring theme is that when the underlying system is "sufficiently hyperbolic", many of these ratios turn out to be bounded away from 0 and infinity. The resulting uniform ratio bounds are central to the study of the measure of maximal entropy and other equilibrium measures. In these lectures, I will survey some of these bounds, their proofs, and their consequences for existence, uniqueness, and other results in thermodynamic formalism. In particular, I will describe recent work that extends this story beyond the classical setting of continuous systems on compact spaces. This includes geodesic flows on non-compact negatively curved manifolds under a strong positive recurrence condition, using a version of Bowen's specification property, and also includes Sinai billiard maps using growth-fragmentation lemmas and a Hausdorff measure construction.
By Vaughn Climenhaga
By Vaughn Climenhaga
