American Institute of Mathematical Sciences

Advanced
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.  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.  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.   Google Scholar

[4]

Sébastien Gouëzel and Carlangelo Liverani, Banach spaces adapted to Anosov systems,, Ergodic Theory Dynam. Systems, 26 (2006), 189.  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, 54 (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.  doi: 10.2307/2160348.  Google Scholar

[7]

Morris W. Hirsch, Charles Pugh and Michael Shub, "Invariant Manifolds,'', Lecture Notes in Mathematics, 583 (1977).   Google Scholar

[8]

Lars Hörmander, "The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis,'', Second edition, 256 (1990).   Google Scholar

[9]

Carlangelo Liverani, On contact Anosov flows,, Ann. of Math. (2), 159 (2004), 1275.  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.   Google Scholar

[11]

Roger D. Nussbaum, The radius of the essential spectrum,, Duke Math. J., 37 (1970), 473.  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.  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.  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.   Google Scholar

[4]

Sébastien Gouëzel and Carlangelo Liverani, Banach spaces adapted to Anosov systems,, Ergodic Theory Dynam. Systems, 26 (2006), 189.  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, 54 (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.  doi: 10.2307/2160348.  Google Scholar

[7]

Morris W. Hirsch, Charles Pugh and Michael Shub, "Invariant Manifolds,'', Lecture Notes in Mathematics, 583 (1977).   Google Scholar

[8]

Lars Hörmander, "The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis,'', Second edition, 256 (1990).   Google Scholar

[9]

Carlangelo Liverani, On contact Anosov flows,, Ann. of Math. (2), 159 (2004), 1275.  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.   Google Scholar

[11]

Roger D. Nussbaum, The radius of the essential spectrum,, Duke Math. J., 37 (1970), 473.  doi: 10.1215/S0012-7094-70-03759-2.  Google Scholar

[1]

Martin Brokate, Pavel Krejčí. Weak differentiability of scalar hysteresis operators. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2405-2421. doi: 10.3934/dcds.2015.35.2405
[2]

Frédéric Naud. The Ruelle spectrum of generic transfer operators. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2521-2531. doi: 10.3934/dcds.2012.32.2521
[3]

Frédéric Naud. Birkhoff cones, symbolic dynamics and spectrum of transfer operators. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 581-598. doi: 10.3934/dcds.2004.11.581
[4]

Damien Thomine. A spectral gap for transfer operators of piecewise expanding maps. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 917-944. doi: 10.3934/dcds.2011.30.917
[5]

Vesselin Petkov, Luchezar Stoyanov. Ruelle transfer operators with two complex parameters and applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6413-6451. doi: 10.3934/dcds.2016077
[6]

Gary Froyland, Cecilia González-Tokman, Anthony Quas. Detecting isolated spectrum of transfer and Koopman operators with Fourier analytic tools. Journal of Computational Dynamics, 2014, 1 (2) : 249-278. doi: 10.3934/jcd.2014.1.249
[7]

Doug Hensley. Continued fractions, Cantor sets, Hausdorff dimension, and transfer operators and their analytic extension. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2417-2436. doi: 10.3934/dcds.2012.32.2417
[8]

Eberhard Bänsch, Steffen Basting, Rolf Krahl. Numerical simulation of two-phase flows with heat and mass transfer. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2325-2347. doi: 10.3934/dcds.2015.35.2325
[9]

Elena Kartashova. Nonlinear resonances of water waves. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 607-621. doi: 10.3934/dcdsb.2009.12.607
[10]

Darryl D. Holm, Cornelia Vizman. Dual pairs in resonances. Journal of Geometric Mechanics, 2012, 4 (3) : 297-311. doi: 10.3934/jgm.2012.4.297
[11]

Maciej Zworski. A remark on inverse problems for resonances. Inverse Problems & Imaging, 2007, 1 (1) : 225-227. doi: 10.3934/ipi.2007.1.225
[12]

Kazuyuki Yagasaki. Degenerate resonances in forced oscillators. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 423-438. doi: 10.3934/dcdsb.2003.3.423
[13]

Wael Bahsoun, Paweł Góra. SRB measures for certain Markov processes. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 17-37. doi: 10.3934/dcds.2011.30.17
[14]

Dieter Bothe, Jan Prüss. Modeling and analysis of reactive multi-component two-phase flows with mass transfer and phase transition the isothermal incompressible case. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 673-696. doi: 10.3934/dcdss.2017034
[15]

Hiroaki Yoshimura, Jerrold E. Marsden. Dirac cotangent bundle reduction. Journal of Geometric Mechanics, 2009, 1 (1) : 87-158. doi: 10.3934/jgm.2009.1.87
[16]

Indranil Biswas, Georg Schumacher, Lin Weng. Deligne pairing and determinant bundle. Electronic Research Announcements, 2011, 18: 91-96. doi: 10.3934/era.2011.18.91
[17]

Todd Kapitula, Björn Sandstede. Eigenvalues and resonances using the Evans function. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 857-869. doi: 10.3934/dcds.2004.10.857
[18]

Misha Guysinsky, Boris Hasselblatt, Victoria Rayskin. Differentiability of the Hartman--Grobman linearization. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 979-984. doi: 10.3934/dcds.2003.9.979
[19]

Eleonora Catsigeras, Heber Enrich. SRB measures of certain almost hyperbolic diffeomorphisms with a tangency. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 177-202. doi: 10.3934/dcds.2001.7.177
[20]

Carlangelo Liverani. Fredholm determinants, Anosov maps and Ruelle resonances. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1203-1215. doi: 10.3934/dcds.2005.13.1203

2018 Impact Factor: 0.295

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences