Journal of Modern Dynamics (JMD)

Robustly invariant sets in fiber contracting bundle flows

Pages: 255 - 267, Issue 2, June 2013      doi:10.3934/jmd.2013.7.255

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Oliver Butterley - Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria (email)
Carlangelo Liverani - Dipartimento di Matematica, II Università di Roma (Tor Vergata), Via della Ricerca Scientifica, 00133 Roma, Italy (email)

Abstract: We provide abstract conditions which imply the existence of a robustly invariant neighborhood of a global section of a fiber bundle flow. We then apply such a result to the bundle flow generated by an Anosov flow when the fiber 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 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 Smooth Anosov flows: correlation spectra and stability [2] by the present authors.

Keywords:  Bundle flows, Transfer operators, resonances, differentiability of SRB.
Mathematics Subject Classification:  Primary: 37C30; Secondary: 37D30, 37M25.

Received: October 2012;      Available Online: September 2013.