Thermodynamic Formalism Beyond Uniformly Hyperbolic Systems PDF Download

One of the primary tools used to study hyperbolic and expanding dynamical systems are equilibrium measures. These are probability measures that are invariant under the dynamical system, and which maximize the sum of the metric entropy of the dynamics and the integral of a particular function (this maximum is taken over all invariant probability measures). Within the equilibrium measures are the Sinai-Ruelle-Bowen (SRB) measures. These are equilibrium measures [mu] that are "physical" in the sense that for Lebesgue-almost every point x in the phase space X and every continuous real-valued function [phi] on X , the Birkhoff time average is equal to the space of the function [phi]. It is known that if a surface diffeomorphism admits an SRB measure, then this measure is unique. Furthermore, hyperbolic maps admitting singularities have at most countably many ergodic SRB measures. In this dissertation, we consider the SRB measures and thermodynamic properties of three classes of dynamical systems: almost Anosov toral diffeomorphisms; smooth realizations of pseudo-Anosov maps; and singular hyperbolic attractors. Almost Anosov and pseudo-Anosov maps are surface diffeomorphisms admitting a finite number of singular fixed points, where the differential of the map slows to the identity. We develop a thermodynamic formalism for both classes of diffeomorphisms. In particular, we use a Young tower construction to prove the existence and uniqueness of equilibrium measures for a collection of non-Hölder continuous geometric potentials. For both almost and pseudo-Anosov maps, this family of equilibrium measures includes a unique SRB measure and a measure of maximal entropy. This measure of maximal entropy also has exponential decay of correlations and satisfies the Central Limit Theorem. Singular hyperbolic attractors are the invariant supports of physical measures for dissipative hyperbolic systems. Examples of singular hyperbolic attractors include the geometric Lorenz attractor (as well as related Lorenz-type systems), and the Belykh attractor. We establish easily verifiable sufficient conditions guaranteeing that the number of ergodic SRB measures is at most finite, and provide examples and nonexamples showing that the conditions are necessary.