Cocycles of isometries and denseness of domination

TL;DR

This paper investigates when linear cocycles over minimal diffeomorphisms can be approximated by cocycles with dominated splittings, showing that this depends only on their homotopy class for fiber dimension at least 3.

Contribution

It establishes a homotopy-based criterion for approximating cocycles by those with dominated splittings over minimal dynamics, extending understanding of cocycle structure.

Findings

Approximation depends solely on homotopy class for fiber dimension ≥ 3.

Introduces a general theorem on almost invariant sections for fiberwise isometries.

Utilizes a quantitative homotopy result to achieve the main theorem.

Abstract

We consider the problem of approximating a linear cocycle (or, more generally, a vector bundle automorphism) over a fixed base dynamics by another cocycle admitting a dominated splitting. We prove that the possibility of doing so depends only on the homotopy class of the cocycle, provided that the base dynamics is a minimal diffeomorphism and the fiber dimension is least 3. This result is obtained by means of a general theorem on the existence of almost invariant sections for fiberwise isometries of bundles with compact fibers and finite fundamental group. The main novelty of the proofs is the use of a quantitative homotopy result due to Calder, Siegel, and Williams.