Drew Ash (Davidson), Ergodic Theory and Dynamical Systems Semianr

Title:  Odometers, Substitutions, and Bounded Topological Speedups

Abstract: In dynamics, when we study a transformation of a space T:X->X, we often take the point of view that the action of T represents the passage of a unit of time, and thus by studying (X,T) we are studying how the space X evolves over time. In keeping with this view point, we can define a “speedup” of T –  a transformation S:X->X  which is obtained by applying a positive power p(x) of T to each point x. While speedups have their origin in the measurable (ergodic) category, we will exclusively look at speedups in the topological category. In this talk, we will discuss bounded topological speedups of minimal Cantor systems. That is, we consider a fixed minimal Cantor system (X,T), and look at homeomorphisms of the form x 􏰀→ Tp(x)(x) where p is a bounded, positive, integer valued function. This talk will be rooted in two classical minimal Cantor systems: odometers and primitive substitution systems. The talk will culminate by showing a bounded topological speedup of an odometer is a conjugate odometer. Should time allow, we will discuss the primitive substitution case. This talk is based on joint work with Lori Alvin and Nic Ormes. Further, all relevant definitions will be provided during the talk.