Ergodic Theory and Dynamical Systems

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

a1 Department of Mathematics, University of Groningen, PO Box 800, 9700 AV, Groningen, The Netherlands (e-mail:
a2 Eurandom, Technical University of Eindhoven, PO Box 513, 5600 MB, Eindhoven, The Netherlands (e-mail:


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.

(Received April 16 2001)
(Revised December 14 2001)