Robustly invariant sets in fibre contracting bundle flows

TL;DR

This paper establishes conditions for the existence of robustly invariant neighborhoods in fibre bundle flows, applies these to Anosov flows with jet spaces, and uses the results to correct a previous mistake in the literature.

Contribution

It introduces abstract conditions for robust invariance in fibre bundle flows and applies them to Anosov flows, also correcting a prior error in related research.

Findings

Existence of robustly invariant neighborhoods around global sections.

Application to Anosov flows with jet space fibers.

Correction of a mistake in previous work on smooth Anosov flows.

Abstract

We provide abstract conditions which imply the existence of a robustly invariant neighbourhood of a global section of a fibre bundle flow. We then apply such a result to the bundle flow generated by an Anosov flow when the fibre is the space of jets (which are described by local manifolds). As a consequence we obtain sets of manifolds (e.g. approximations of stable manifolds) that are left invariant, {\bf for all} negative times, by the flow and its small perturbations. Finally, we show that the latter result can be used to easily fix a mistake recently uncovered in the paper {\em Smooth Anosov flows: correlation spectra and stability}, \cite{BuL}, by the present authors.