On simultaneous linearization of certain commuting nearly integrable diffeomorphisms of the cylinder

TL;DR

This paper proves that certain commuting smooth diffeomorphisms of the cylinder can be simultaneously linearized under specific conditions, extending the understanding of local rigidity in dynamical systems.

Contribution

It introduces a KAM-based method for simultaneous linearization of commuting diffeomorphisms near integrable maps, with necessary conditions and applications to rigidity of -actions.

Findings

Simultaneous linearization is possible under intersection and semi-conjugacy conditions.

Provides examples demonstrating the necessity of these conditions.

Establishes local rigidity results for commuting twist maps on the cylinder.

Abstract

Let $\mathcal{F}$ and $\mathcal{K}$ be commuting $C^\infty$ diffeomorphisms of the cylinder $\mathbb{T}\times\mathbb{R}$ that are, respectively, close to $\mathcal{F}_0 (x, y)=(x+\omega(y), y)$ and $T_\alpha (x, y)=(x+\alpha, y)$, where $\omega(y)$ is non-degenerate and $\alpha$ is Diophantine. Using the KAM iterative scheme for the group action we show that $\mathcal{F}$ and $\mathcal{K}$ are simultaneously $C^\infty$-linearizable if $\mathcal{F}$ has the intersection property (including the exact symplectic maps) and $\mathcal{K}$ satisfies a semi-conjugacy condition. We also provide examples showing necessity of these conditions. As a consequence, we get local rigidity of certain class of $\mathbb{Z}^2$-actions on the cylinder, generated by commuting twist maps.