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

Let $(X,d)$ be a compact metric space, $f:X \mapsto X$ be a continuous map with the specification property, and $φ: X \mapsto \IR$ 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 \[ \{x \in X : \lim_{n \ra \infty} \frac{1}{n} \sum_{i = 0}^{n-1} φ(f^i (x)) = α\}. \] 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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