Random subshifts of finite type
Abstract
Let $X$ be an irreducible shift of finite type (SFT) of positive entropy, and let $B_n(X)$ be its set of words of length $n$. Define a random subset $\omega$ of $B_n(X)$ by independently choosing each word from $B_n(X)$ with some probability $\alpha$. Let $X_{\omega}$ be the (random) SFT built from the set $\omega$. For each $0\leq \alpha \leq1$ and $n$ tending to infinity, we compute the limit of the likelihood that $X_{\omega}$ is empty, as well as the limiting distribution of entropy for $X_{\omega}$. For $\alpha$ near 1 and $n$ tending to infinity, we show that the likelihood that $X_{\omega}$ contains a unique irreducible component of positive entropy converges exponentially to 1. These results are obtained by studying certain sequences of random directed graphs. This version of "random SFT" differs significantly from a previous notion by the same name, which has appeared in the context of random dynamical systems and bundled dynamical systems.
 Publication:

arXiv eprints
 Pub Date:
 June 2010
 arXiv:
 arXiv:1006.1325
 Bibcode:
 2010arXiv1006.1325M
 Keywords:

 Mathematics  Probability;
 Mathematics  Dynamical Systems
 EPrint:
 Published in at http://dx.doi.org/10.1214/10AOP636 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)