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A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions. (English) Zbl 0985.37040

Summary: The authors establish the complete multifractal formalism for equilibrium measures for Hölder continuous conformal expanding maps and expanding Markov Moran-like geometric constructions. Examples include Markov maps of an interval, beta transformations of an interval, rational maps with hyperbolic Julia sets, and conformal toral endomorphisms. We also construct a Hölder continuous homeomorphism of a compact metric space with an ergodic invariant measure of positive entropy for which the dimension spectrum is not convex, and hence the multifractal formalism fails.

MSC:

37F15 Expanding holomorphic maps; hyperbolicity; structural stability of holomorphic dynamical systems
28A78 Hausdorff and packing measures
37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
37C45 Dimension theory of smooth dynamical systems
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
28A80 Fractals
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