Strong Approximations of Shifts and the Characteristic Measures Problem

2023/07/12 by Van Cyr, Cyr, Van, Bryna Kra +3
Computer Science · Mathematics · #37A15 #37B10 #68R15 #Advanced Mathematical Modeling in Engineering #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2307.06451

openalex publication_date 2023/07/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Every symbolic system supports a Borel measure that is invariant under the shift, but it is not known if every such systems supports a measure that is invariant under all of its automorphisms; known as a characteristic measure. We give sufficient conditions to find a characteristic measure, additionally showing when it can be taken to be a measure of maximal entropy. The class of systems to which these sufficient conditions apply is large, containing a dense G_δ set in the space of all shifts on a given alphabet, and is also large in the sense that it is closed under taking factors. We also investigate natural systems to which these sufficient conditions apply.

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