Bounded topological speedups

ABSTRACT

This paper explores the range of bounded speedups in the topological category. Bounded speedups represent both a strengthening of topological speedups as first defined by the second author. 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.

KEYWORDS:

• Topological speedups
• odometers
• substitutions
• Kakutani–Rokhlin towers
• entropy

2010 Mathematics Subject Classification:

• Primary 37B05
• Secondary 37A20
• 37A25
• 37B10
• 37B40
• 54H20

Acknowledgment

The authors would thank the referee for many helpful comments on early drafts of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.