Physical-like Measures Coincide with Invariant Measures Supported on Chain Recurrent Classes

2019/07/20 by Xueting Tian, Tian, Xueting
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1907.08755

openalex publication_date 2019/07/20 · openalex created_date 2019/07/30 · openalex updated_date 2026/07/28

Abstract

For C⁰ generic continuous maps or homeomorphisms on compact Riemannian manifold, we prove that (1) the space of physical-like measures coincides with the set of invariant measures supported on chain recurrent classes, (2) every point in the base space is typical (that is, for any point in the base space, its empirical measures are contained in the space of physical-like measures) and (3) there is a subset of strongly regular set with Lebesgue zero measure but has infinite topological entropy. Moreover, some comparison between C⁰ generic systems and C⁰ conservative generic systems are discussed.

Citations

• Variational principles and mixed multifractal spectra
• ω-Limit sets for Axiom A diffeomorphisms
• Sets of “Non-typical” points have full topological entropy and full Hausdorff dimension
• Topological entropy for noncompact sets
• Irregular sets, the 𝛽-transformation and the almost specification property

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