KAM-renormalization and Herman rings for 2D maps

TL;DR

This paper extends a renormalization framework to 2D maps, demonstrating that Herman rings with certain rotation numbers persist under small perturbations, revealing stability properties of these structures.

Contribution

It introduces an extension of the renormalization horseshoe to 2D maps and proves the persistence of Herman rings with bounded type rotation numbers under perturbations.

Findings

Herman rings survive on a codimension one set of parameters.

The renormalization horseshoe is extended to 2D maps.

Herman rings with bounded type rotation numbers are stable under small perturbations.

Abstract

In this note, we extend the renormalization horseshoe we have recently constructed with N. Goncharuk for analytic diffeomorphisms of the circle to their small two-dimensional perturbations. As one consequence, Herman rings with rotation numbers of bounded type survive on a codimension one set of parameters under small two-dimensional perturbations.