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Center for Nonlinear Studies |
In the theory of dynamical systems the so-called thermodynamic formalism, originally developed by theoretical physicists, has become a powerful tool for studying geometric and ergodic-theoretical aspects. One main object in this theory is the topological pressure (related to the free energy functional), i.e. a particular functional on the space of observables, that encodes several important quantities of the underlying dynamical system. For example, the pressure is related to entropy, Lyapunov exponents, dimension, multifractal spectra, natural invariant measures, etc. For hyperbolic systems Rufus Bowen and David Ruelle established in their pioneer works deep connections between topological pressure and periodic points, Hausdorff dimenison and the characterization of attractors. The main purpose of this work is to generalize some of these results to the case of nonuniformly hyperbolic systems.