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Some results on recurrence and entropy
Pavlov, Ronald L., Jr.

2007, Doctor of Philosophy, Ohio State University, Mathematics.

This thesis is comprised primarily of two separate portions. In the first portion, we exhibit, for any sparse enough increasing sequence of integers {p_n}, a totally minimal, totally uniquely ergodic, and topologically mixing system (X,T) and a continuous function f on X for which the ergodic averages along {p_n} fail to converge for a residual set in X, answering negatively an open question of Bergelson. We also construct a totally minimal, totally uniquely ergodic, and topologically mixing system (X',T') and x' a point in X' so that x' is not a limit point of {T^(p_n)(x')}.

In the second portion, we study perturbations of multidimensional shifts of finite type. Given any Z^d shift of finite type X for d>1 and any word w in the language of X, denote by X_w the set of elements of X in which w does not appear. If X satisfies a uniform mixing condition called strong irreducibility, we obtain exponential upper and lower bounds on the difference in the topological entropies of X and X_w dependent only on the size of w. This result generalizes a result of Lind about Z shifts of finite type.

Vitaly Bergelson (Advisor)
175 p.

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