Uniformly quasiconformal partially hyperbolic systems
[Systèmes partiellement hyperboliques uniformément quasi conformes]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 5, pp. 1085-1127.

Nous étudions les perturbations lisses préservant le volume de l'application temps-un du flot géodésique ψt d'une variété riemannienne fermée de dimension au moins égale à trois et de courbure négative constante. Nous montrons que pour une telle perturbation, les exposants de Lyapunov extrémaux relativement au volume coïncident à la fois dans les sous-espaces stables et instables si et seulement si cette perturbation se plonge comme temps-un d'un flot lisse préservant le volume et dont les orbites sont conjuguées de manière lisse à celles de ψt. Nos techniques s'appliquent plus généralement pour donner une classification essentiellement complète des difféomorphismes lisses, partiellement hyperboliques préservant le volume et vérifient une condition de quasi-conformalité uniforme le long de leurs fibrés stables et instables qui, soit possèdent un feuilletage central compact avec une holonomie triviale, soit sont obtenus comme perturbations de l'application temps-un d'un flot d'Anosov.

We study smooth volume-preserving perturbations of the time-1 map of the geodesic flow ψt of a closed Riemannian manifold of dimension at least three with constant negative curvature. We show that such a perturbation has equal extremal Lyapunov exponents with respect to volume within both the stable and unstable bundles if and only if it embeds as the time-1 map of a smooth volume-preserving flow that is smoothly orbit equivalent to ψt. Our techniques apply more generally to give an essentially complete classification of smooth, volume-preserving partially hyperbolic diffeomorphisms which satisfy a uniform quasiconformality condition on their stable and unstable bundles and have either compact center foliation with trivial holonomy or are obtained as perturbations of the time-1 map of an Anosov flow.

DOI : 10.24033/asens.2372
Classification : 37D30, 37C85, 30C65.
Keywords: Partially hyperbolic diffeomorphisms, Lyapunov exponent, quasiconformal mapping, rigidity.
Mot clés : Difféomorphismes partiellement hyperboliques, exposant de Liapounov, application quasi conforme, rigidité.
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Butler, Clark; Xu, Disheng. Uniformly quasiconformal partially hyperbolic systems. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 5, pp. 1085-1127. doi : 10.24033/asens.2372. http://archive.numdam.org/articles/10.24033/asens.2372/

Avila, A.; Santamaria, J.; Viana, M. Holonomy invariance: rough regularity and applications to Lyapunov exponents, Astérisque, Volume 358 (2013), pp. 13-74 (ISBN: 978-2-85629-778-0, ISSN: 0303-1179) | Numdam | MR | Zbl

Abdenur, F.; Viana, M. Flavors of partial hyperbolicity (preprint http://w3.impa.br/~viana/out/flavors.pdf )

Avila, A.; Viana, M.; Wilkinson, A. Absolute continuity, Lyapunov exponents and rigidity I: geodesic flows, J. Eur. Math. Soc. (JEMS), Volume 17 (2015), pp. 1435-1462 (ISSN: 1435-9855) | DOI | MR | Zbl

Bohnet, D.; Bonatti, C. Partially hyperbolic diffeomorphisms with a uniformly compact center foliation: the quotient dynamics, Ergodic Theory and Dynamical Systems, Volume 36 (2016), pp. 1067-1105 | DOI | MR | Zbl

Besson, G.; Courtois, G.; Gallot, S. Entropies et rigidités des espaces localement symétriques de courbure strictement négative, Geom. Funct. Anal., Volume 5 (1995), pp. 731-799 (ISSN: 1016-443X) | DOI | MR | Zbl

Brin, M.; Karcher, H. Frame flows on manifolds with pinched negative curvature, Compos. math., Volume 52 (1984), pp. 275-297 (ISSN: 0010-437X) | Numdam | MR | Zbl

Bohnet, D. Partially hyperbolic diffeomorphisms with a compact center foliation with finite holonomy. Ph.D. Thesis (2011)

Burns, K.; Pollicott, M. Stable ergodicity and frame flows, Geom. Dedicata, Volume 98 (2003), pp. 189-210 (ISSN: 0046-5755) | DOI | MR | Zbl

Brin, M.; Stuck, G., Cambridge Univ. Press, Cambridge, 2002, 240 pages (ISBN: 0-521-80841-3) | DOI | MR | Zbl

Butler, C. Rigidity of equality of Lyapunov exponents for geodesic flows, J. Diff. Geom., Volume 109 (2018), pp. 39-79 | MR | Zbl

Bonatti, C.; Wilkinson, A. Transitive partially hyperbolic diffeomorphisms on 3-manifolds, Topology, Volume 44 (2005), pp. 475-508 | DOI | MR | Zbl

Burns, K.; Wilkinson, A. Dynamical coherence and center bunching, Discrete Contin. Dyn. Syst, Volume 22 (2008), pp. 89-100 | DOI | MR | Zbl

Burns, K.; Wilkinson, A. On the ergodicity of partially hyperbolic systems, Ann. of Math., Volume 171 (2010), pp. 451-489 (ISSN: 0003-486X) | DOI | MR | Zbl

Carrasco, P. D. Compact dynamical foliations, Ergodic theory and dynamical systems, Volume 35 (2015), pp. 2474-2498 | DOI | MR | Zbl

Eberlein, P. B., Chicago Lectures in Mathematics, University of Chicago Press, Chicago, 1996, 449 pages (ISBN: 0-226-18197-9; 0-226-18198-7) | MR | Zbl

Epstein, D. B. Foliations with all leaves compact, Annales de l'institut Fourier, Volume 26 (1976), pp. 265-282 | DOI | Numdam | MR | Zbl

Epstein, D. B.; Vogt, E. A counterexample to the periodic orbit conjecture in codimension 3, Annals of Mathematics, Volume 108 (1978), pp. 539-552 | DOI | MR | Zbl

Fang, Y. Smooth rigidity of uniformly quasiconformal Anosov flows, Ergodic Theory Dynam. Systems, Volume 24 (2004), pp. 1937-1959 (ISSN: 0143-3857) | DOI | MR | Zbl

Fang, Y. On the rigidity of quasiconformal Anosov flows, Ergodic Theory Dynam. Systems, Volume 27 (2007), pp. 1773-1802 (ISSN: 0143-3857) | DOI | MR | Zbl

Fang, Y. Quasiconformal Anosov flows and quasisymmetric rigidity of Hamenstädt distances, Discrete Contin. Dyn. Syst., Volume 34 (2014), pp. 3471-3483 (ISSN: 1078-0947) | DOI | MR | Zbl

Gehring, F. W. Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc., Volume 103 (1962), pp. 353-393 (ISSN: 0002-9947) | DOI | MR | Zbl

Golʼdsheĭd, I. Y.; Margulis, G. A. Lyapunov exponents of a product of random matrices, Uspekhi Mat. Nauk, Volume 44 (1989), pp. 13-60 (ISSN: 0042-1316) | DOI | MR | Zbl

Gogolev, A. Partially hyperbolic diffeomorphisms with compact center foliations (preprint arXiv:1104.5464 ) | MR | Zbl

Gromov, M., Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) (Ann. of Math. Stud.), Volume 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 183-213 | DOI | MR | Zbl

Hertz, F. R.; Hertz, M. R.; Ures, R. A non-dynamically coherent example on 𝕋 3 , Annales de l'Institut Henri Poincare (C) Non Linear Analysis (2015) | Numdam | MR | Zbl

Hirsch, M. W.; Pugh, C.; Shub, M., Lecture Notes in Math., 583, Springer, Berlin-New York, 1977, 149 pages | MR | Zbl

Journé, J.-L. A regularity lemma for functions of several variables, Rev. Mat. Iberoamericana, Volume 4 (1988), pp. 187-193 (ISSN: 0213-2230) | DOI | MR | Zbl

Kanai, M. Differential-geometric studies on dynamics of geodesic and frame flows, Japan. J. Math. (N.S.), Volume 19 (1993), pp. 1-30 (ISSN: 0289-2316) | DOI | MR | Zbl

Kingman, J. F. C. The ergodic theory of subadditive stochastic processes, J. Roy. Statist. Soc. Ser. B, Volume 30 (1968), pp. 499-510 (ISSN: 0035-9246) | DOI | MR | Zbl

Katok, A.; Kononenko, A. Cocycles' stability for partially hyperbolic systems, Math. Res. Lett., Volume 3 (1996), pp. 191-210 (ISSN: 1073-2780) | DOI | MR | Zbl

Kalinin, B.; Sadovskaya, V. Linear cocycles over hyperbolic systems and criteria of conformality, J. Mod. Dyn., Volume 4 (2010), pp. 419-441 (ISSN: 1930-5311) | DOI | MR | Zbl

Kalinin, B.; Sadovskaya, V. Cocycles with one exponent over partially hyperbolic systems, Geom. Dedicata, Volume 167 (2013), pp. 167-188 (ISSN: 0046-5755) | DOI | MR | Zbl

Livšic, A. N.; Sinaĭ, J. G. Invariant measures that are compatible with smoothness for transitive C-systems, Dokl. Akad. Nauk SSSR, Volume 207 (1972), pp. 1039-1041 (ISSN: 0002-3264) | MR

Pugh, C.; Shub, M.; Wilkinson, A. Hölder foliations, Duke Math. J., Volume 86 (1997), pp. 517-546 (ISSN: 0012-7094) | DOI | MR | Zbl

Rohlin, V. A. On the fundamental ideas of measure theory, Mat. Sbornik N.S., Volume 25(67) (1949), pp. 107-150 | MR

Sadovskaya, V. On uniformly quasiconformal Anosov systems, Math. Res. Lett., Volume 12 (2005), pp. 425-441 (ISSN: 1073-2780) | DOI | MR | Zbl

Smale, S. Differentiable dynamical systems, Bull. Amer. Math. Soc., Volume 73 (1967), pp. 747-817 (ISSN: 0002-9904) | DOI | MR | Zbl

Sullivan, D. A counterexample to the periodic orbit conjecture, Publ. math. IHÉS, Volume 46 (1976), pp. 5-14 | DOI | Numdam | MR | Zbl

Sullivan, D. Quasiconformal homeomorphisms in dynamics, topology, and geometry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI (1987), pp. 1216-1228 | MR | Zbl

Shub, M.; Wilkinson, A. Pathological foliations and removable zero exponents, Invent. math., Volume 139 (2000), pp. 495-508 (ISSN: 0020-9910) | DOI | MR | Zbl

Tukia, P. On quasiconformal groups, J. Analyse Math., Volume 46 (1986), pp. 318-346 (ISSN: 0021-7670) | DOI | MR | Zbl

Väisälä, J., Lecture Notes in Math., 229, Springer, Berlin-New York, 1971, 144 pages | MR | Zbl

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