Orders of Accumulation of Entropy and Random Subshifts of Finite Type

Title of dissertation: ORDERS OF ACCUMULATION OF ENTROPY AND RANDOM SUBSHIFTS OF FINITE TYPE Kevin McGoff, Doctor of Philosophy, 2011 Dissertation directed by: Professor Mike Boyle Department of Mathematics The first portion of this dissertation concerns orders of accumulation of entropy. For a continuous map T of a compact metrizable space X with finite topological entropy, the order of accumulation of entropy of T is a countable ordinal that arises in the context of entropy structure and symbolic extensions. We show that every countable ordinal is realized as the order of accumulation of some dynamical system. Our proof relies on the functional analysis of metrizable Choquet simplices and a realization theorem of Downarowicz and Serafin. Further, ifM is a metrizable Choquet simplex, we bound the ordinals that appear as the order of accumulation of entropy of a dynamical system whose simplex of invariant measures is affinely homeomorphic to M . These bounds are given in terms of the Cantor-Bendixson rank of ex(M), the closure of the extreme points of M , and the relative CantorBendixson rank of ex(M) with respect to ex(M). We also address the optimality of these bounds. Given any compact manifoldM and any countable ordinal α, we also construct a continuous, surjective self-map of M having order of accumulation of entropy α. If the dimension of M is at least 2, then the map can be chosen to be a homeomorphism. The realization theorem of Downarowicz and Serafin produces dynamical systems on the Cantor set; by contrast, our constructions work on any manifold and provide a more direct dynamical method of obtaining systems with prescribed entropy properties. Next we consider random subshifts of finite type. Let X be an irreducible shift of finite type (SFT) of positive entropy, and let Bn(X) be its set of words of length n. Define a random subset ω of Bn(X) by independently choosing each word from Bn(X) with some probability α. Let Xω be the (random) SFT built from the set ω. For each 0 ≤ α ≤ 1 and n tending to infinity, we compute the limit of the likelihood that Xω is empty, as well as the limiting distribution of entropy for Xω. For α near 1 and n tending to infinity, we show that the likelihood that Xω 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. ORDERS OF ACCUMULATION OF ENTROPY AND RANDOM SUBSHIFTS OF FINITE TYPE