Random $\mathbb{Z}^d$-shifts of finite type

Pages: 287 - 330,
Volume 10,
2016 doi:10.3934/jmd.2016.10.287

Kevin McGoff - Department of Mathematics and Statistics, The University of North Carolina at Charlotte, 9201 University City Blvd., Charlotte, NC 28223, United States (email)

Ronnie Pavlov - Department of Mathematics, University of Denver, 2280 S. Vine St., Denver, CO 80208, United States (email)

Abstract:
In this work we consider an ensemble of random $\mathbb{Z}^d$-shifts of finite type ($\mathbb{Z}^d$-SFTs) and prove several results concerning the behavior of typical systems with respect to emptiness, entropy, and periodic points. These results generalize statements made in [26] regarding the case $d=1$.

Let $\mathcal{A}$ be a finite set, and let $d \geq 1$. For $n$ in $\mathbb{N}$ and $\alpha$ in $[0,1]$, define a random subset $\omega$ of $\mathcal{A}^{[1,n]^d}$ by independently including each pattern in $\mathcal{A}^{[1,n]^d}$ with probability $\alpha$. Let $X_{\omega}$ be the (random) $\mathbb{Z}^d$-SFT built from the set $\omega$. For each $\alpha \in [0,1]$ and $n$ tending to infinity, we compute the limit of the probability that $X_{\omega}$ is empty, as well as the limiting distribution of entropy of $X_{\omega}$. Furthermore, we show that the probability of obtaining a nonempty system without periodic points tends to zero.

For $d>1$, the class of $\mathbb{Z}^d$-SFTs is known to contain strikingly different behavior than is possible within the class of $\mathbb{Z}$-SFTs. Nonetheless, the results of this work suggest a new heuristic: typical $\mathbb{Z}^d$-SFTs have similar properties to their $\mathbb{Z}$-SFT counterparts.

Keywords: Symbolic dynamics, subshifts of finite type, topological entropy, random dynamical systems.

Mathematics Subject Classification: Primary: 37B50; Secondary: 37B10, 37H99.

Received: August 2014;
Revised:
January 2016;
Available Online: July 2016.

References