Birkhoff spectra for one-dimensional maps with some hyperbolicity

2008/03/11 by Yong Moo Chung, Chung, Yong Moo
Mathematics · Physics and Astronomy · #37C45 #Chaos control and synchronization #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #math.DS #msc:37C45

paper · pdf · doi:10.48550/arxiv.0803.1522

21 pages

arxiv created 2008/03/11 · openalex publication_date 2008/03/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the multifractal analysis for smooth dynamical systems in dimension one. It is characterized the Hausdorff dimension of the level set obtained from the Birkhoff averages of a continuous function by the local dimensions of hyperbolic measures for a topologically mixing C² map modelled by an abstract dynamical system. A characterization which corresponds to above is also given for the ergodic basins of invariant probability measures. And it is shown that the complement of the set of quasi-regular points carries full Hausdorff dimension.

Citations

• Sets of “Non-typical” points have full topological entropy and full Hausdorff dimension
• Dimension, entropy and Lyapunov exponents
• Recurrence times and large deviations

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• BIRKHOFF SPECTRA FOR ONE-DIMENSIONAL MAPS WITH SOME HYPERBOLICITY
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• Multifractal analysis of weak Gibbs measures for non-uniformly expanding C¹ maps
• Multifractal analysis of the divergence points of Birkhoff averages in beta-dynamical systems