Robustly invariant sets in fibre contracting bundle flows

Oliver Butterley, Carlangelo Liverani

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.