Topological pressure via saddle points
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- by Katrin Gelfert and Christian Wolf PDF
- Trans. Amer. Math. Soc. 360 (2008), 545-561 Request permission
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 $f|\Lambda$) 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.References
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Additional Information
- 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
- MR Author ID: 687384
- Email: gelfert@pks.mpg.de
- Christian Wolf
- Affiliation: Department of Mathematics, Wichita State University, Wichita, Kansas 67260
- MR Author ID: 673329
- Email: cwolf@math.wichita.edu
- 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.
- © Copyright 2007 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 360 (2008), 545-561
- MSC (2000): Primary 37D25, 37D35; Secondary 37C25, 37C40, 37C45
- DOI: https://doi.org/10.1090/S0002-9947-07-04407-8
- MathSciNet review: 2342015