Topological pressure via saddle points
Authors:
Katrin Gelfert and Christian Wolf
Journal:
Trans. Amer. Math. Soc. 360 (2008), 545561
MSC (2000):
Primary 37D25, 37D35; Secondary 37C25, 37C40, 37C45
Published electronically:
July 23, 2007
MathSciNet review:
2342015
Fulltext PDF Free Access
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Abstract: Let be a compact locally maximal invariant set of a  diffeomorphism on a smooth Riemannian manifold . In this paper we study the topological pressure (with respect to the dynamical system ) 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 is entirely determined by the values of on the saddle points of in . Moreover, it is enough to consider saddle points with ``large'' Lyapunov exponents. We also introduce a version of the pressure for certain noncontinuous potentials and establish several variational inequalities for it. Finally, we deduce relations between expansion and escape rates and the dimension of . Our results generalize several wellknown results to certain nonuniformly hyperbolic systems.
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Additional Information
Katrin Gelfert
Affiliation:
MaxPlanckInstitut für Physik komplexer Systeme, Nöthnitzer Str. 38, D01187 Dresden, Germany – and – Institut für Physik, TU Chemnitz, D09107 Chemnitz, Germany
Email:
gelfert@pks.mpg.de
Christian Wolf
Affiliation:
Department of Mathematics, Wichita State University, Wichita, Kansas 67260
Email:
cwolf@math.wichita.edu
DOI:
http://dx.doi.org/10.1090/S0002994707044078
PII:
S 00029947(07)044078
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. EPS0236913 and matching support from the State of Kansas through Kansas Technology Enterprise Corporation.
Article copyright:
© Copyright 2007
American Mathematical Society
