Transitivity of Infinite-Dimensional Extensions of Anosov Diffeomorphisms

TL;DR

This paper extends the concept of transitivity in dynamical systems to infinite-dimensional settings, specifically for extensions of Anosov diffeomorphisms, and demonstrates the existence and limitations of such transitive cocycles.

Contribution

It generalizes finite-dimensional results to infinite dimensions, showing how to construct transitive cocycles and highlighting the failure of stable transitivity in infinite-dimensional cases.

Findings

Any Holder cocycle satisfying an obstruction induces topologically transitive extension.

Constructed cocycles satisfy conditions for any Anosov diffeomorphism.

No infinite-dimensional stably transitive cocycles exist under various metrics.

Abstract

We consider extensions of Anosov diffeomorphisms of an infranilmanifold by the real vector space R^{\omega}. Our main result, based on the analogous theorem in finite dimensions proven by Nitica and Pollicott, is that any Holder cocycle satisfying an obvious obstruction induces a topologically transitive extension (topologically weak mixing, in fact). We show how to construct cocycles satisfying these conditions for any Anosov diffeomorphism, and then observe that unlike the finite dimensional case, where cocycles satisfying the obstruction are C^0-stably transitive, there can be no infinite-dimensional stably transitive cocycles, with respect to several spaces and metrics of cocycles.