Robustly invariant sets in fiber contracting bundle flows

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April 2013, 7(2): 255-267. doi: 10.3934/jmd.2013.7.255

Robustly invariant sets in fiber contracting bundle flows

Oliver Butterley ^1, and Carlangelo Liverani ^2,

1.	Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
2.	Dipartimento di Matematica, II Università di Roma (Tor Vergata), Via della Ricerca Scientifica, 00133 Roma

Received October 2012 Published September 2013

Abstract
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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: Transfer operators, resonances, differentiability of SRB., Bundle flows.

Mathematics Subject Classification: Primary: 37C30; Secondary: 37D30, 37M2.

Citation: Oliver Butterley, Carlangelo Liverani. Robustly invariant sets in fiber contracting bundle flows. Journal of Modern Dynamics, 2013, 7 (2) : 255-267. doi: 10.3934/jmd.2013.7.255

References:

[1]	Viviane Baladi and Carlangelo Liverani, Exponential decay of correlations for piecewise cone hyperbolic contact flows, Communications in Mathematical Physics, 314 (2012), 689-773. doi: 10.1007/s00220-012-1538-4.
[2]	Oliver Butterley and Carlangelo Liverani, Smooth Anosov flows: Correlation spectra and stability, J. Mod. Dyn., 1 (2007), 301-322. doi: 10.3934/jmd.2007.1.301.
[3]	Paolo Giulietti, Carlangelo Liverani and Mark Pollicott, Anosov flows and dynamical zeta functions, Annals of Mathematics, 178 (2013), 687-773.
[4]	Sébastien Gouëzel and Carlangelo Liverani, Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems, 26 (2006), 189-217. doi: 10.1017/S0143385705000374.
[5]	Anatole Katok and Boris Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,'' With a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.
[6]	Hubert Hennion, Sur un théorème spectral et son application aux noyaux Lipchitziens, Proceedings of the American Mathematical Society, 118 (1993), 627-634. doi: 10.2307/2160348.
[7]	Morris W. Hirsch, Charles Pugh and Michael Shub, "Invariant Manifolds,'' Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.
[8]	Lars Hörmander, "The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis,'' Second edition, Grundlehren der Mathematischen Wissenschaften, 256, Springer-Verlag, Berlin, 1990.
[9]	Carlangelo Liverani, On contact Anosov flows, Ann. of Math. (2), 159 (2004), 1275-1312. doi: 10.4007/annals.2004.159.1275.
[10]	John N. Mather, Characterization of Anosov diffeomorphisms, Nederl. Akad. Wetensch. Proc. Ser. A 71 = Indag. Math., 30 (1968), 479-483.
[11]	Roger D. Nussbaum, The radius of the essential spectrum, Duke Math. J., 37 (1970), 473-478. doi: 10.1215/S0012-7094-70-03759-2.

show all references

References:

[1]	Viviane Baladi and Carlangelo Liverani, Exponential decay of correlations for piecewise cone hyperbolic contact flows, Communications in Mathematical Physics, 314 (2012), 689-773. doi: 10.1007/s00220-012-1538-4.
[2]	Oliver Butterley and Carlangelo Liverani, Smooth Anosov flows: Correlation spectra and stability, J. Mod. Dyn., 1 (2007), 301-322. doi: 10.3934/jmd.2007.1.301.
[3]	Paolo Giulietti, Carlangelo Liverani and Mark Pollicott, Anosov flows and dynamical zeta functions, Annals of Mathematics, 178 (2013), 687-773.
[4]	Sébastien Gouëzel and Carlangelo Liverani, Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems, 26 (2006), 189-217. doi: 10.1017/S0143385705000374.
[5]	Anatole Katok and Boris Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,'' With a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.
[6]	Hubert Hennion, Sur un théorème spectral et son application aux noyaux Lipchitziens, Proceedings of the American Mathematical Society, 118 (1993), 627-634. doi: 10.2307/2160348.
[7]	Morris W. Hirsch, Charles Pugh and Michael Shub, "Invariant Manifolds,'' Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.
[8]	Lars Hörmander, "The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis,'' Second edition, Grundlehren der Mathematischen Wissenschaften, 256, Springer-Verlag, Berlin, 1990.
[9]	Carlangelo Liverani, On contact Anosov flows, Ann. of Math. (2), 159 (2004), 1275-1312. doi: 10.4007/annals.2004.159.1275.
[10]	John N. Mather, Characterization of Anosov diffeomorphisms, Nederl. Akad. Wetensch. Proc. Ser. A 71 = Indag. Math., 30 (1968), 479-483.
[11]	Roger D. Nussbaum, The radius of the essential spectrum, Duke Math. J., 37 (1970), 473-478. doi: 10.1215/S0012-7094-70-03759-2.

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