2005/09/27 by Katrin Gelfert, Christian Wolf, Gelfert, Katrin +1
Mathematics · Physics and Astronomy · #37C25 #37C45 #37D25 #37D35 #Chaos control and synchronization #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.math/0509630
openalex publication_date 2005/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Λ be a compact locally maximal invariant set of a C2-diffeomorphism f:M→ M on a smooth Riemannian manifold M. In this paper we study the topological pressure P\rm top(ϕ) (with respect to the dynamical system f|Λ) for a wide class of Hölder continuous potentials and analyze its relation to dynamical, as well as geometrical, properties of the system. We show that under a mild nonuniform hyperbolicity assumption the topological pressure of ϕ is entirely determined by the values of ϕ on the saddle points of f in Λ. Moreover, it is enough to consider saddle points with ``large'' Lyapunov exponents. We also introduce a version of the pressure for certain non-continuous potentials and establish several variational inequalities for it. Finally, we deduce relations between expansion and escape rates and the dimension of Λ. Our results generalize several well-known results to certain non-uniformly hyperbolic systems.