Bounded Topological Speedups

TL;DR

This paper investigates bounded speedups in topological dynamics, showing they preserve certain classical systems, alter entropy bounds, and are generally not conjugate to original systems, thus expanding understanding of their structural effects.

Contribution

It introduces the concept of bounded speedups in topological systems, demonstrating their preservation of classical structures and providing entropy bounds and calculations.

Findings

Bounded speedups of odometers are conjugate odometers.

Bounded speedups of primitive substitutions remain primitive but are not conjugate.

Bounds and calculations for topological entropy of bounded speedups.

Abstract

This paper explores the range of bounded speedups in the topological category. Bounded speedups represent both a strengthening of topological speedups as defined in [A 16] and a generalization of powers of a transformation. Here we show that bounded speedups preserve the structure of two classical minimal Cantor systems. Specifically, a minimal bounded speedup of an odometer is a conjugate odometer, and a minimal bounded speedup of a primitive substitution is again a primitive substitution, though it is never conjugate to the original substitution system. Further, we give bounds on the topological entropy of bounded speedups, and in special cases we compute the topological entropy of bounded speedups.