Topological Speedups.

摘要

Given a dynamical system T : X → X one can define a speedup of (X, T) as another dynamical system conjugate to S : X → X where S(x) = T p(x)(x) for some function p : X → Z+. In 1985 Arnoux, Ornstein, and Weiss showed that any aperiodic measure preserving system is isomorphic to a speedup of any ergodic measure preserving system. In this thesis we study speedups in the topological category. Specifically, we consider minimal homeomorphisms on Cantor spaces. Our main theorem gives conditions on when one such system is a speedup of another. Moreover, the main theorem serves as a topological analogue of the Arnoux, Ornstein, and Weiss speedup theorem, as well as a one-sided analogue of Giordano, Putnam, and Skau's characterization of orbit equivalence. Further, this thesis explores the special case of speedups when the p function is bounded. In this case, we provide bounds on the entropy of bounded speedups.