Linear cocycles over hyperbolic systems and criteria of conformality

TL;DR

This paper investigates Holder continuous linear cocycles over transitive Anosov diffeomorphisms, establishing conditions for conformality and isometry based on periodic data, and analyzing invariant sub-bundles and structures.

Contribution

It provides new criteria linking periodic data to cocycle conformality and isometry, with results on invariant sub-bundles under pinching conditions.

Findings

Periodic data determines cocycle conformality

Existence of invariant sub-bundles under pinching conditions

Cocycles are conformal or isometric if periodic maps are

Abstract

In this paper we study Holder continuous linear cocycles over transitive Anosov diffeomorphisms. Under various conditions of relative pinching we establish properties including existence and continuity of measurable invariant sub-bundles and conformal structures. We use these results to obtain criteria for cocycles to be isometric or conformal in terms of their periodic data. We show that if the return maps at the periodic points are, in a sense, conformal or isometric then so is the cocycle itself with respect to a Holder continuous Riemannian metric.