Transitivity of Infinite-Dimensional Extensions of Anosov Diffeomorphisms

Zev Rosengarten, Asaf Reich

Published 2012-09-10Version 1

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.