Counterexamples to a rigidity conjecture

TL;DR

This paper presents counterexamples to a rigidity conjecture by Khanin, showing that certain non-rigid diffeomorphisms and flows defy the expected smooth conjugacy under specific conditions, highlighting the complexity of rigidity phenomena.

Contribution

The authors construct explicit counterexamples to Khanin's rigidity conjecture for diffeomorphisms and flows, emphasizing the role of invariant distributions in rigidity questions.

Findings

Counterexamples of non-rigid diffeomorphisms on the 2-torus.

Examples of topologically conjugate but not $C^1$ conjugate flows.

Indication that invariant distributions influence rigidity properties.

Abstract

We discuss several counterexamples to a rigidity conjecture of K. Khanin, which states that under some quantitative condition on non-existence of periodic orbits, $C^0$ conjugacy implies $C^1$ (even $C^\infty$) conjugacy. We construct examples of non-rigid diffeomorphisms on the $2$-torus, which satisfy the assumptions of Khanin's (but not of Krikorian's) conjecture. We also construct examples of flows which are topologically conjugate, but not $C^1$ conjugate, in contradiction to a natural generalization of the conjecture to flows. These latter examples are based on results on solutions of the cohomological equation and suggest that the structure of the space of invariant distributions has to be taken into account in rigidity questions.