Dominated splitting from constant periodic data and global rigidity of Anosov automorphisms

TL;DR

This paper proves that certain hyperbolic systems with constant or near-constant periodic data exhibit dominated splittings and rigidity properties, extending to some non-linear Anosov diffeomorphisms close to automorphisms.

Contribution

It establishes the existence of dominated splittings from periodic data and demonstrates global rigidity for a class of Anosov automorphisms and nearby non-linear systems.

Findings

Dominated splitting from constant periodic data in hyperbolic systems.

Global periodic data rigidity for generic Anosov automorphisms.

Rigidity results extend to systems with near-constant periodic data.

Abstract

We show that a $\mathrm{GL}(d,\mathbb{R})$ cocycle over a hyperbolic system with constant periodic data has a dominated splitting whenever the periodic data indicates it should. This implies global periodic data rigidity of generic Anosov automorphisms of $\mathbb{T}^d$. Further, our approach also works when the periodic data is narrow, that is, sufficiently close to constant. We can show global periodic data rigidity for certain non-linear Anosov diffeomorphisms in a neighborhood of an irreducible Anosov automorphism with simple spectrum.