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Let $X$ be a compact space, $f\colon X \to X$ a continuous map, and $\Lambda \subset X$ be any $f$-invariant subset. Assume that there exists a nested family of subsets $\{\Lambda_l\}_{l \geq 1}$ that exhaust $\Lambda$, that is $\Lambda_l \subset\Lambda_{l+1}$ and $\Lambda =\bigcup_{l \geq 1} \Lambda_l$. Assume that the potential $\varphi \colon X \to \mathbb{R}$ is continuous on the closure of each $\Lambda_l$ but not necessarily continuous on $\Lambda$. We define the topological pressure of $\varphi$ on $\Lambda$. This definition is shown to have a corresponding variational principle. We apply the topological pressure and variational principle to systems with non-zero Lyapunov exponents, countable Markov shifts, and unimodal maps.


This is the author's manuscript. The version of record is available from the publisher at Copyright © 2006 Cambridge University Press. All rights reserved.

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