Renormalization of circle maps and smoothness of Arnold tongues

TL;DR

This paper investigates the renormalization operator on a Banach manifold with critical circle maps and circle diffeomorphisms, leading to new results on the smoothness of irrational Arnold tongues.

Contribution

It introduces a novel analysis of the renormalization operator on a specialized Banach manifold and applies it to establish smoothness properties of Arnold tongues.

Findings

Proves smoothness of irrational Arnold tongues.

Analyzes the global behavior of the renormalization operator.

Constructs a Banach manifold with specific boundary and interior properties.

Abstract

We study the global behavior of the renormalization operator on a specially constructed Banach manifold that has cubic critical circle maps on its boundary and circle diffeomorphisms in its interior. As an application, we prove results on smoothness of irrational Arnold tongues.