A variational principle for topological pressure for certain non-compact sets

2009/09/04 by Daniel Thompson
Mathematics · Physics and Astronomy · #Mathematical Dynamics and Fractals #Statistical Mechanics and Entropy #Geometric Analysis and Curvature Flows

paper · doi:10.1112/jlms/jdp041

Abstract

Let (X, d) be a compact metric space, let f:X ↦ X be a continuous map with the specification property and let φ: X ↦ ℝ be a continuous function. We prove a variational principle for topological pressure (in the sense of Pesin and Pitskel) for non-compact sets of the form x ∈ X : lim n → ∞ 1 n ∑ i − 0 n − 1 φ ( f i ( x ) ) = α Analogous results were previously known for topological entropy. As an application, we prove multifractal analysis results for the entropy spectrum of a suspension flow over a continuous map with specification and the dimension spectrum of certain non-uniformly expanding interval maps.

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