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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Topological pressure via saddle points

Authors: Katrin Gelfert and Christian Wolf
Journal: Trans. Amer. Math. Soc. 360 (2008), 545-561
MSC (2000): Primary 37D25, 37D35; Secondary 37C25, 37C40, 37C45
Published electronically: July 23, 2007
MathSciNet review: 2342015
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Abstract: Let $ \Lambda$ be a compact locally maximal invariant set of a $ C^2$- diffeomorphism $ f:M\to M$ on a smooth Riemannian manifold $ M$. In this paper we study the topological pressure $ P_{\mathrm{top}}(\varphi)$ (with respect to the dynamical system $ f\vert\Lambda$) 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 non-uniform hyperbolicity assumption the topological pressure of $ \varphi$ is entirely determined by the values of $ \varphi$ on the saddle points of $ f$ in $ \Lambda$. 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 $ \Lambda$. Our results generalize several well-known results to certain non-uniformly hyperbolic systems.

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Katrin Gelfert
Affiliation: Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, D-01187 Dresden, Germany – and – Institut für Physik, TU Chemnitz, D-09107 Chemnitz, Germany

Christian Wolf
Affiliation: Department of Mathematics, Wichita State University, Wichita, Kansas 67260

Keywords: Topological pressure, $C^2$-diffeomorphism, saddle points, invariant measures, Hausdorff dimension
Received by editor(s): September 8, 2006
Published electronically: July 23, 2007
Additional Notes: The research of the first author was supported by the Deutsche Forschungsgemeinschaft. The research of the second author was supported in part by the National Science Foundation under Grant No. EPS-0236913 and matching support from the State of Kansas through Kansas Technology Enterprise Corporation.
Article copyright: © Copyright 2007 American Mathematical Society

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