Deformation and perturbative rigidity near de la Llave examples

TL;DR

This paper investigates the rigidity and deformation properties of certain Anosov diffeomorphisms on the four-torus, showing that while they are not globally conjugate to linear models, they exhibit local rigidity near specific examples.

Contribution

It establishes local deformation and rigidity results for generic diffeomorphisms close to de la Llave's examples, highlighting their local exceptional behavior.

Findings

De la Llave's examples are not globally conjugate to linear models.

Generic diffeomorphisms near these examples exhibit local rigidity.

The examples are 'locally exceptional' in their deformation properties.

Abstract

De la Llave's examples are Anosov diffeomorphisms on the four-torus $\mathbb{T}^4$ with constant Lyapunov spectrum, yet they are not $C^{1}$-conjugate to the linear model or to each other. Nevertheless, we show that such examples are ``locally exceptional'': we prove deformation and local rigidity for generic diffeomorphisms in proximity of de la Llave's examples.