$C^1$ Robust Rigidity for Bi-critical Circle Maps

TL;DR

This paper proves that bi-critical circle maps with matching signatures and exponentially converging renormalizations in the $C^2$-topology are actually $C^1$ conjugate, establishing a rigidity result.

Contribution

It establishes a $C^1$ rigidity result for bi-critical circle maps under specific topological and renormalization convergence conditions.

Findings

Bi-critical circle maps with same signatures are $C^1$ conjugate.

Exponential convergence of renormalizations in $C^2$ implies $C^1$ conjugacy.

Provides conditions under which topological conjugacy implies smooth conjugacy.

Abstract

We prove that two topologically conjugate bi-critical circle maps whose signatures are the same, and whose renormalizations converge together exponentially fast in the $C^2$-topology, are $C^1$ conjugate.