Generic Points of shift-Invariant Measures in the Countable Symbolic Space

2016/01/29 by Aihua Fan, Mingtian Li, Fan, Ai-hua +3
Mathematics · Physics and Astronomy · #11J70 #37B10 #37C45 #Chaos control and synchronization #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1601.08019

openalex publication_date 2016/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We are concerned with sets of generic points for shift-invariant measures in the countable symbolic space. We measure the sizes of the sets by the Billingsley-Hausdorff dimensions defined by Gibbs measures. It is shown that the dimension of such a set is given by a variational principle involving the convergence exponent of the Gibbs measure and the relative entropy dimension of the Gibbs measure with respect to the invariant measure. This variational principle is different from that of the case of finite symbols, where the convergent exponent is zero and is not involved. An application is given to a class of expanding interval dynamical systems.

Citations

• Topological entropy for noncompact sets

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