Differentiable conjugacies for one-dimensional maps

TL;DR

This paper explores the concept of differentiable conjugacies in one-dimensional maps, detailing techniques, recent results, and their implications for classifying dynamical systems beyond topological equivalence.

Contribution

It introduces methods for defining differentiable conjugacies in bifurcations and chaotic maps, leading to new normal forms and recognition criteria for such conjugacies.

Findings

Differentiable conjugacies exist for certain standard bifurcations.

Closed-form expressions are available for some chaotic maps.

Constraints for recognizing differentiable conjugacies are identified.

Abstract

Differentiable conjugacies link dynamical systems that share properties such as the stability multipliers of corresponding orbits. It provides a stronger classification than topological conjugacy, which only requires qualitative similarity. We describe some of the techniques and recent results that allow differentiable conjugacies to be defined for standard bifurcations, and explain how this leads to a new class of normal forms. Closed-form expressions for differentiable conjugacies exist between some chaotic maps, and we describe some of the constraints that make it possible to recognise when such conjugacies arise. This paper focuses on the consequences of the existence of differentiable conjugacies rather than the conjugacy classes themselves.