The natural measure of a symbolic dynamical system

2013/08/13 by Hu, Wen-Guei, Lin, Song-Sun
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1308.2996

Abstract

This study investigates the natural or intrinsic measure of a symbolic dynamical system Σ. The measure μ([i₁,i₂,...,i_n]) of a pattern [i₁,i₂,...,i_n] in Σ is an asymptotic ratio of [i₁,i₂,...,i_n], which arises in all patterns of length n within very long patterns, such that in a typical long pattern, the pattern [i₁,i₂,...,i_n] appears with frequency μ([i₁,i₂,...,i_n]). When Σ=Σ(A) is a shift of finite type and A is an irreducible N× N non-negative matrix, the measure μ is the Parry measure. μ is ergodic with maximum entropy. The result holds for sofic shift G=(G,L), which is irreducible. The result can be extended to Σ(A), where A is a countably infinite matrix that is irreducible, aperiodic and positive recurrent. By using the Krieger cover, the natural measure of a general shift space is studied in the way of a countably infinite state of sofic shift, including context free shift. The Perron-Frobenius Theorem for non-negative matrices plays an essential role in this study.

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