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On the variational principle for the topological entropy of certain non-compact sets

Published online by Cambridge University Press:  24 January 2003

FLORIS TAKENS
Affiliation:
Department of Mathematics, University of Groningen, PO Box 800, 9700 AV, Groningen, The Netherlands (e-mail: f.takens@math.rug.nl)
EVGENY VERBITSKIY
Affiliation:
Eurandom, Technical University of Eindhoven, PO Box 513, 5600 MB, Eindhoven, The Netherlands (e-mail: e.verbitskiy@tue.nl)

Abstract

For a continuous transformation f of a compact metric space (X,d) and any continuous function \phi on X we consider sets of the form

K_{\alpha} =\bigg\{x\in X:\lim_{n\to\infty} \frac 1n \sum_{i=0}^{n-1} \phi( f^i(x))=\alpha \bigg\},\quad\alpha\in\R.

For transformations satisfying the specification property we prove the following Variational Principle

h_{\rm top}(f,K_{\alpha}) = \sup\bigg( h_\mu(f): \mu\text{ is invariant and } \int\phi \,d\mu=\alpha \bigg),

where h_{\rm top}(f,\cdot) is the topological entropy of non-compact sets. Using this result we are able to obtain a complete description of the multifractal spectrum for Lyapunov exponents of the so-called Manneville–Pomeau map, which is an interval map with an indifferent fixed point.

We also consider multi-dimensional multifractal spectra and establish a contraction principle.

Type
Research Article
Copyright
2003 Cambridge University Press

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