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A thermodynamic definition of topological pressure for non-compact sets

Published online by Cambridge University Press:  26 March 2010

DANIEL J. THOMPSON
DANIEL J. THOMPSON*
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, State College, PA 16802, USA (email: [email protected])
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Abstract

We give a new definition of topological pressure for arbitrary (non-compact, non-invariant) Borel subsets of metric spaces. This new quantity is defined via a suitable variational principle, leading to an alternative definition of an equilibrium state. We study the properties of this new quantity and compare it with existing notions of topological pressure. We are particularly interested in the situation when the ambient metric space is assumed to be compact. We motivate our definition by applying it to some interesting examples, including the level sets of the pointwise Lyapunov exponent for the Manneville–Pomeau family of maps.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 2 , April 2011 , pp. 527 - 547
Copyright
Copyright © Cambridge University Press 2010

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