On expanding maps and topological conjugacy

Abstract

This paper deals with expanding maps on compact spaces and the applicability of Banach's fixed point theorem to the conjugacy problem. We provide a criterion for a smooth map on a manifold to be expanding on a compact invariant set. In general, this criterion is only sufficient but not necessary. But in two important cases, expanding maps on the circle and hyperbolic rational maps on the Riemann sphere, we prove equivalence. Furthermore, we provide a similar criterion for an interval map to be piecewise expanding, and for two interval maps to be conjugate. As an example we use our expansiveness criterion to give a short and simple proof of hyperbolicity for the logistic maps with parameter α>4. Moreover, we prove that the logistic maps and a special tent map are conjugate, and show piecewise conjugacy for another family of unimodal maps.

Keywords:

• Expanding maps
• Hyperbolicity
• Topological conjugacy
• Banach's fixed point

Keywords:

• 37C15
• 37E05
• 37F15

Acknowledgements

My gratitude is to Prof. Bernd Aulbach for introducing me to the beautiful theory of dynamical systems.