TY - JOUR

T1 - On the variational principle for the topological entropy of certain non-compact sets

AU - Takens, F.

AU - Verbitskiy, E.A.

PY - 2003

Y1 - 2003

N2 - 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.

AB - 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.

U2 - 10.1017/S0143385702000913

DO - 10.1017/S0143385702000913

M3 - Article

SN - 0143-3857

VL - 23

SP - 317

EP - 348

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

IS - 1

ER -