On the asymptotic measure of periodic subsystems of finite type in symbolic dynamics

TL;DR

This paper investigates the asymptotic decay of measures of certain periodic subsystems in symbolic dynamics, extending previous results to include cases where the subsystem is irreducible and periodic, and provides explicit examples of differing behaviors.

Contribution

It extends existing asymptotic measure results to periodic subsystems in symbolic dynamics, highlighting differences from the aperiodic case and providing explicit examples.

Findings

Asymptotic measure decay differs in periodic subsystems compared to aperiodic ones.

Extension of previous results to irreducible and periodic subsystems.

Explicit example demonstrating different asymptotic behavior.

Abstract

Let $\Delta\subsetneq\V$ be a proper subset of the vertices $\V$ of the defining graph of an aperiodic shift of finite type $(\Sigma_{A}^{+},\S)$. Let $\Delta_{n}$ be the union of cylinders in $\Sigma_{A}^{+}$ corresponding to the points $x$ for which the first $n$-symbols of $x$ belong to $\Delta$ and let $\mu$ be an equilibrium state of a H\"older potential $\phi$ on $\Sigma_{A}^{+}$. We know that $\mu(\Delta_{n})$ converges to zero as $n$ diverges. We study the asymptotic behaviour of $\mu(\Delta_{n})$ and compare it with the pressure of the restriction of $\phi$ to $\Sigma_{\Delta}$. The present paper extends some results in \cite{CCC} to the case when $\Sigma_{\Delta}$ is irreducible and periodic. We show an explicit example where the asymptotic behaviour differs from the aperiodic case.