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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

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.  Google Scholar

[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.  Google Scholar

[3]

Paolo Giulietti, Carlangelo Liverani and Mark Pollicott, Anosov flows and dynamical zeta functions, Annals of Mathematics, 178 (2013), 687-773. Google Scholar

[4]

Sébastien Gouëzel and Carlangelo Liverani, Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems, 26 (2006), 189-217. doi: 10.1017/S0143385705000374.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

Morris W. Hirsch, Charles Pugh and Michael Shub, "Invariant Manifolds,'' Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[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.  Google Scholar

[9]

Carlangelo Liverani, On contact Anosov flows, Ann. of Math. (2), 159 (2004), 1275-1312. doi: 10.4007/annals.2004.159.1275.  Google Scholar

[10]

John N. Mather, Characterization of Anosov diffeomorphisms, Nederl. Akad. Wetensch. Proc. Ser. A 71 = Indag. Math., 30 (1968), 479-483.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

Paolo Giulietti, Carlangelo Liverani and Mark Pollicott, Anosov flows and dynamical zeta functions, Annals of Mathematics, 178 (2013), 687-773. Google Scholar

[4]

Sébastien Gouëzel and Carlangelo Liverani, Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems, 26 (2006), 189-217. doi: 10.1017/S0143385705000374.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

Morris W. Hirsch, Charles Pugh and Michael Shub, "Invariant Manifolds,'' Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[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.  Google Scholar

[9]

Carlangelo Liverani, On contact Anosov flows, Ann. of Math. (2), 159 (2004), 1275-1312. doi: 10.4007/annals.2004.159.1275.  Google Scholar

[10]

John N. Mather, Characterization of Anosov diffeomorphisms, Nederl. Akad. Wetensch. Proc. Ser. A 71 = Indag. Math., 30 (1968), 479-483.  Google Scholar

[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.  Google Scholar

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