Renormalization of bicritical circle maps

TL;DR

This paper proves that exponential convergence in renormalization for a broad class of bicritical circle maps leads to their smooth conjugacy, extending previous results from single to bicritical cases and implying rigidity for certain analytic maps.

Contribution

It extends the renormalization approach to bicritical circle maps, establishing smooth conjugacy results previously known for single critical point maps.

Findings

Exponential convergence implies smooth conjugacy for bicritical circle maps.

Established $C^{1+eta}$ rigidity for real-analytic bicritical maps with bounded type rotation number.

Adapted existing proofs to handle two critical points in the renormalization framework.

Abstract

A general ansatz in Renormalization Theory, already established in many important situations, states that exponential convergence of renormalization orbits implies that topological conjugacies are actually smooth (when restricted to the attractors of the original systems). In this paper we establish this principle for a large class of bicritical circle maps, which are $C^3$ circle homeomorphisms with irrational rotation number and exactly two (non-flat) critical points. The proof presented here is an adaptation, to the bicritical setting, of the one given by de Faria and de Melo for the case of a single critical point. When combined with some recent papers, our main theorem implies $C^{1+\alpha}$ rigidity for real-analytic bicritical circle maps with rotation number of bounded type.