The Rigidity Conjecture

TL;DR

This paper discusses the concept of rigidity in dynamical systems, exploring when topological properties determine geometric structure and proposing a conjecture on the organization of topological and rigidity classes.

Contribution

It introduces a conjecture that explains the organization of topological classes into rigidity classes in dynamical systems.

Findings

Rigidity holds for many classical dynamical systems under mild conditions.

Recent results show rigidity does not always hold for more general systems.

The paper proposes a conjecture on the organization of topological and rigidity classes.

Abstract

A central question in dynamics is whether the topology of a system determines its geometry. This is known as rigidity. Under mild topological conditions rigidity holds for many classical cases, including: Kleinian groups, circle diffeomorphisms, unimodal interval maps, critical circle maps, and circle maps with a break point. More recent developments show that under similar topological conditions, rigidity does not hold for slightly more general systems. In this paper we state a conjecture which describes how topological classes are organized into rigidity classes.