Topological rigidity of unfoldings of resonant diffeomorphisms

TL;DR

This paper establishes that topological conjugacies between generic unfoldings of resonant complex diffeomorphisms are necessarily holomorphic or anti-holomorphic on the unperturbed parameter, revealing a form of topological rigidity.

Contribution

It proves a rigidity theorem for conjugacies of unfoldings of resonant diffeomorphisms, showing they are holomorphic or anti-holomorphic under generic conditions and characterizing their behavior.

Findings

Conjugacies are holomorphic or anti-holomorphic on the unperturbed parameter.

Genericity is necessary for the rigidity result.

Conjugacies are always real analytic outside the origin.

Abstract

We prove that a topological homeomorphism conjugating two generic 1-parameter unfoldings of 1-variable complex analytic resonant diffeomorphisms is holomorphic or anti-holomorphic by restriction to the unperturbed parameter. We provide examples that show that the genericity hypothesis is necessary. Moreover we characterize the possible behavior of conjugacies for the unperturbed parameter in the general case. In particular they are always real analytic outside of the origin. We describe the structure of the limits of orbits when we approach the unperturbed parameter. The proof of the rigidity results is based on the study of the action of a topological conjugation on the limits of orbits.