Orders of accumulation of entropy and random subshifts of finite type

dc.contributor.advisorBoyle, McBlaine Men_US
dc.contributor.authorMcGoff, Kevin Alexanderen_US
dc.contributor.departmentMathematicsen_US
dc.contributor.publisherDigital Repository at the University of Marylanden_US
dc.contributor.publisherUniversity of Maryland (College Park, Md.)en_US
dc.date.accessioned2011-07-07T05:32:23Z
dc.date.available2011-07-07T05:32:23Z
dc.date.issued2011en_US
dc.description.abstractFor 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, if M 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 F, the closure of the extreme points of M, and the relative Cantor-Bendixson rank of F with respect to the extreme points of M. We address the optimality of these bounds. Given any compact manifold M and any countable ordinal alpha, we construct a continuous, surjective self-map of M having order of accumulation of entropy alpha. 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 with its set of words of length n denoted B_n(X). Define a random subset E of B_n(X) by independently choosing each word from B_n(X) with some probability alpha. Let X_E be the (random) SFT built from the set E. For each alpha in [0,1] and n tending to infinity, we compute the limit of the likelihood that X_E; is empty, as well as the limiting distribution of entropy for X_E. For alpha near 1 and n tending to infinity, we show that the likelihood that X_E contains a unique irreducible component of positive entropy converges exponentially to 1.en_US
dc.identifier.urihttp://hdl.handle.net/1903/11628
dc.subject.pqcontrolledMathematicsen_US
dc.subject.pquncontrolledentropyen_US
dc.subject.pquncontrolledsubshifts of finite typeen_US
dc.subject.pquncontrolledsymbolic extensionsen_US
dc.titleOrders of accumulation of entropy and random subshifts of finite typeen_US
dc.typeDissertationen_US

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