The natural measure of a symbolic dynamical system

Abstract

This study investigates the natural or intrinsic measure of a symbolic dynamical system $\Sigma$. The measure $\mu([i_{1},i_{2},...,i_{n}])$ of a pattern $[i_{1},i_{2},...,i_{n}]$ in $\Sigma$ is an asymptotic ratio of $[i_{1},i_{2},...,i_{n}]$, which arises in all patterns of length $n$ within very long patterns, such that in a typical long pattern, the pattern $[i_{1},i_{2},...,i_{n}]$ appears with frequency $\mu([i_{1},i_{2},...,i_{n}])$. When $\Sigma=\Sigma(A)$ is a shift of finite type and $A$ is an irreducible $N\times N$ non-negative matrix, the measure $\mu$ is the Parry measure. $\mu$ is ergodic with maximum entropy. The result holds for sofic shift $\mathcal{G}=(G,\mathcal{L})$, which is irreducible. The result can be extended to $\Sigma(A)$, 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.