Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T13:19:11.343Z Has data issue: false hasContentIssue false
  • English
  • Français

Partial hyperbolicity and classification: a survey

Published online by Cambridge University Press:  19 September 2016

ANDY HAMMERLINDL  and
RAFAEL POTRIE
ANDY HAMMERLINDL
Affiliation:
School of Mathematical Sciences, Monash University, Victoria 3800, Australia email andy.hammerlindl@monash.edu
RAFAEL POTRIE
Affiliation:
CMAT, Facultad de Ciencias, Universidad de la República, Uruguay email rpotrie@cmat.edu.uy
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper surveys recent results on classifying partially hyperbolic diffeomorphisms. This includes the construction of branching foliations and leaf conjugacies on three-dimensional manifolds with solvable fundamental group. Classification results in higher-dimensional settings are also discussed. The paper concludes with an overview of the construction of new partially hyperbolic examples derived from Anosov flows.

Type
Survey Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 2 , April 2018 , pp. 401 - 443
Copyright
© Cambridge University Press, 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anosov, D.. Geodesic flows on closed Riemannian manifolds of negative curvature. Trudy Mat. Inst. Steklov. 90 (1967), 3–210.Google Scholar
Araújo, V. and Pacifico, M. J.. Three Dimensional Flows (Ergebnisse der Mathematik und ihrer Grenzgebiete) . Springer, Berlin, 2010.CrossRefGoogle Scholar
Barbot, T.. Flots d’Anosov sur les variétés graphées au sens de Waldhausen. Ann. Inst. Fourier (Grenoble) 46 (1996), 1451–1517.Google Scholar
Barbot, T. and Fenley, S.. Pseudo Anosov flows in toroidal manifolds. Geom. Topol. 17 (2013), 1877–1954.CrossRefGoogle Scholar
Beguin, F., Bonatti, C. and Yu, B.. Building Anosov flows on 3-manifolds. Preprint, 2014, arXiv:1408.3951.Google Scholar
Bonatti, C., Díaz, L. and Pujals, E.. A C 1 -generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources. Ann. of Math. (2) 158(2) (2003), 355–418.CrossRefGoogle Scholar
Bonatti, C. and Díaz, L.. Persistent transitive diffeomorphisms. Ann. of Math. (2) 143(2) (1996), 357–396.CrossRefGoogle Scholar
Bonatti, C., Díaz, L. and Viana, M.. Dynamics beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective (Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics III) . Springer, Berlin, 2005.Google Scholar
Bonatti, C., Gogolev, A., Hammerlindl, A. and Potrie, R.. Anomalous partially hyperbolic diffeomorphisms III: Pseudo Anosov mapping classes. In preparation.Google Scholar
Bonatti, C., Gogolev, A. and Potrie, R.. Anomalous partially hyperbolic diffeomorphisms II: Stably ergodic examples. Invent. Math. to appear, doi:10.1007/s00222-016-0663-7 Preprint, 2015,arXiv:1506.07804.Google Scholar
Bohnet, D.. Codimension-1 partially hyperbolic diffeomorphisms with a uniformly compact center foliation. J. Mod. Dyn. 7 (2013), 565–604.Google Scholar
Bonatti, C. and Langevin, R.. Un exemple de flot d’Anosov transitif transverse à une tore et non conjugué à une suspension. Ergod. Th. & Dynam. Sys. 14 (1994), 633–643.Google Scholar
Bonatti, C., Parwani, K. and Potrie, R.. Anomalous partially hyperbolic diffeomorphisms I: Dynamically coherent examples. Ann. Sci. Éc. Norm. Supér, to appear. Preprint 2014, arXiv:1411.1221.Google Scholar
Bonatti, C. and Wilkinson, A.. Transitive partially hyperbolic diffeomorphisms on 3-manifolds. Topology 44 (2005), 475–508.Google Scholar
Bonatti, C. and Zhang, J.. Transverse foliations on the torus $\mathbb{T}^{2}$ and partially hyperbolic diffeomorphisms on 3-manifolds. Preprint, 2016, arXiv:1602.04355.Google Scholar
Brin, M. and Pesin, Y.. Partially hyperbolic dynamical systems. Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 170–212.Google Scholar
Brin, M.. Topological transitivity of one class of dynamic systems and flows of frames on manifolds of negative curvature. Funct. Anal. Appl. 9 (1975), 8–16.Google Scholar
Brin, M.. On dynamical coherence. Ergod. Th. & Dynam. Sys. 23 (2003), 395–401.Google Scholar
Brin, M., Burago, D. and Ivanov, S.. On partially hyperbolic diffeomorphisms of 3-manifolds with commutative fundamental group. Modern Dynamical Systems and Applications. Cambridge University Press, Cambridge, 2004, pp. 307–312.Google Scholar
Brin, M., Burago, D. and Ivanov, S.. Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus. J. Mod. Dyn. 3 (2009), 1–11.CrossRefGoogle Scholar
Brin, M. and Gromov, M.. On the ergodicity of frame flows. Invent. Math. 60 (1980), 1–8.Google Scholar
Brittenham, M.. Essential laminations in Seifert fibered spaces. Topology 32 (1993), 61–85.Google Scholar
Brin, M. and Manning, A.. Anosov diffeomorphisms with pinched spectrum. Dynamical Systems and Turbulence (Lecture Notes in Mathematics, 898) . Eds. Rand, D. and Young, L.-S.. Springer, Berlin, 1981, pp. 48–53.Google Scholar
Burago, D. and Ivanov, S.. Partially hyperbolic diffeomorphisms of 3-manifolds with abelian fundamental groups. J. Mod. Dyn. 2 (2008), 541–580.Google Scholar
Burns, K., Pugh, C., Shub, M. and Wilkinson, A.. Recent results about stable ergodicity. Proc. Sympos. Amer. Math. Soc. 69 (2001), 327–366.Google Scholar
Burns, K. and Wilkinson, A.. Dynamical coherence and center bunching. Discrete Contin. Dyn. Syst. 22 (2008), 89–100.Google Scholar
Calegari, D.. Foliations and the Geometry of 3-manifolds. Clarendon Press, Oxford, 2008.Google Scholar
Calegari, D. and Dunfield, N.. Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), 149–204.Google Scholar
Carrasco, P.. Compact dynamical foliations. Ergod. Th. & Dynam. Syst. 35(8) (2015), 2474–2498.Google Scholar
Candel, A. and Conlon, L.. Foliations I and II (Graduate studies in Mathematics, 60) . American Mathematical Society, Providence, RI, 2003.Google Scholar
Crovisier, S. and Potrie, R.. Introduction to partially hyperbolic dynamics. Lecture Notes for a Minicourse at ICTP, 2015. Available in the web page of the conference and the authors.Google Scholar
Carrasco, P., Hertz, M. A. R., Hertz, F. R. and Ures, R.. Partially hyperbolic dynamics in dimension 3. Preprint, 2015, arXiv:1501.00932.Google Scholar
Dolgopyat, D. and Pesin, Y.. Every compact manifold carries a completely hyperbolic diffeomorphism. Ergod. Th. & Dynam. Sys. 22 (2002), 409–437.CrossRefGoogle Scholar
Díaz, L., Pujals, E. and Ures, R.. Partial hyperbolicity and robust transitivity. Acta Math. 183 (1999), 1–43.Google Scholar
Fenley, S.. Anosov flows in 3-manifolds. Ann. of Math. (2) 139(1) (1994), 79–115.Google Scholar
Farrell, F. T. and Gogolev, A.. Anosov diffeomorphisms constructed from 𝜋 k (Diff(S n )). J. Topol. 5 (2012), 276–292.Google Scholar
Farrell, F. T. and Gogolev, A.. On bundles that admit fiberwise hyperbolic dynamics. Math. Ann. 364(1) (2016), 401–438.Google Scholar
Farrell, T. F. and Jones, L.. Anosov diffeomorphisms constructed from 𝜋1(Diff(S n )). Topology 17 (1978), 273–282.Google Scholar
Fisher, T., Potrie, R. and Sambarino, M.. Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to Anosov. Math. Z. 278 (2014), 149–168.Google Scholar
Franks, J.. Anosov diffeomorphisms. Global Analysis (Proceedings of Symposia in Pure Mathematics, 14) . Eds. Chern, S. and Smale, S.. American Mathematical Society, Providence, RI, 1970, pp. 61–93.Google Scholar
Franks, J. and Williams, B.. Anomalous Anosov flows. Global Theory of Dynamical Systems (Lecture Notes in Mathematics, 819) . Eds. Nitecki, Z. and Robinson, C.. Springer, Berlin, 1980, pp. 158–174.Google Scholar
Fuller, F. B.. On the surface of section and periodic trajectories. Amer. J. Math. 87 (1965), 473–480.Google Scholar
Ghys, E.. Flots d’Anosov sur les 3-variétés fibrées en cercles. Ergod. Th. & Dynam. Sys. 4 (1984), 67–80.Google Scholar
Gogolev, A.. Partially hyperbolic diffeomorphisms with compact center foliations. J. Mod. Dyn. 5 (2011), 747–767.Google Scholar
Gogolev, A. and Lafont, J. F.. Aspherical products which do not support Anosov diffeomorphisms. Ann. Inst. H. Poincaré, to appear. Preprint, 2015, arXiv:1511.00261.Google Scholar
Goodman, S.. Dehn surgery on Anosov flows. Geometric Dynamics (Rio de Janeiro, 1981) (Lecture Notes in Mathematics, 1007) . Springer, Berlin, 1983, pp. 300–307.Google Scholar
Gogolev, A., Ontaneda, P. and Hertz, F. R.. New partially hyperbolic dynamical systems I. Preprint, 2014, arXiv:1407.7768.Google Scholar
Gogolev, A. and Hertz, F. R.. Manifolds with higher homotopy which do not support Anosov diffeomorphisms. Bull. Lond. Math. Soc. 46(2) (2014), 349–366.Google Scholar
Gourmelon, N. and Potrie, R.. Projectively Anosov diffeomorphisms of surfaces. In preparation.Google Scholar
Grayson, M., Pugh, C. and Shub, M.. Stably ergodic diffeomorphisms. Ann. of Math. (2) 140(2) (1994), 295–330.Google Scholar
Gromov, M.. Three remarks on geodesic dynamics and fundamental group. Preprint SUNY, 1976. Enseign. Math. 46 (2000), 391–402 reprinted.Google Scholar
Gromov, M.. Groups of polynomial growth and expanding maps. Publ. Math. Inst. Hautes Études Sci. 53 (1981), 53–73.Google Scholar
Hammerlindl, A.. Leaf conjugacies in the torus. Ergod. Th. & Dynam. Sys. 33(3) (2013), 896–933.Google Scholar
Hammerlindl, A.. Integrability and Lyapunov exponents. J. Mod. Dyn. 5(1) (2011), 107–122.Google Scholar
Hammerlindl, A.. Dynamics of quasi-isometric foliations. Nonlinearity 25 (2012), 1585–1599.Google Scholar
Hammerlindl, A.. Partial hyperbolicity on 3-dimensional nilmanifolds. Discrete Contin. Dyn. Syst. 33(8) (2013), 3641–3669.Google Scholar
Hammerlindl, A.. Polynomial global product structure. Proc. Amer. Math. Soc. 142(12) (2014), 4297–4303.Google Scholar
Hammerlindl, A.. On expanding foliations. Bull. Braz. Math. Soc. (N.S.) 46(3) (2015), 407–420.Google Scholar
Hammerlindl, A.. Ergodic components of partially hyperbolic systems. Preprint, 2014,arXiv:1409.8002.Google Scholar
Hammerlindl, A. and Potrie, R.. Pointwise partial hyperbolicity in 3-dimensional nilmanifolds. J. Lond. Math. Soc. 89(3) (2014), 853–875.Google Scholar
Hammerlindl, A. and Potrie, R.. Classification of partially hyperbolic diffeomorphisms in 3-manifolds with solvable fundamental group. J. Topol. 8(3) (2015), 842–870.Google Scholar
Hasselblatt, B. and Pesin, Y.. Partially hyperbolic dynamical systems. Handbook of Dynamical Systems 1B. Eds. Hasselblatt, B. and Katok, A. B.. Elsevier, Burlington, MA, 2006, pp. 1–55.Google Scholar
Hammerlindl, A., Potrie, R. and Shannon, M.. Seifert manifolds admitting partially hyperbolic diffeomorphisms. In preparation.Google Scholar
Hatcher, A.. Algebraic Topology. Cambridge University Press, Cambridge, 2002.Google Scholar
Handel, M. and Thurston, W.. Anosov flows on new three manifolds. Invent. Math. 59(2) (1980), 95–103.Google Scholar
Hammerlindl, A. and Ures, R.. Ergodicity and partial hyperbolicity on the 3-torus. Commun. Contemp. Math. 16(4) (2014), 135–158.Google Scholar
Hirsch, M.. Differential Topology (Springer Graduate Texts in Mathematics, 33) . Springer, New York, 1976.Google Scholar
Hirsch, M., Pugh, C. and Shub, M.. Invariant Manifolds (Springer Lecture Notes in Mathematics, 583) . Springer, Berlin, 1977.Google Scholar
Hungerford, T.. Algebra (Springer Graduate Texts in Mathematics, 73) . Reprint of the 1974 original. Springer, Berlin, 1980.Google Scholar
Levitt, G.. Feuilletages des variétés de dimension 3 qui sont fibrés en circles. Comment. Math. Helv. 53 (1978), 572–594.CrossRefGoogle Scholar
Luzzatto, S., Turelli, S. and War, K.. Integrability of dominated decompositions on three-dimensional manifolds. Preprint, 2014, arXiv:1410.8072.Google Scholar
Mañé, R.. Persistent manifolds are normally hyperbolic. Bull. Amer. Math. Soc. 80(1) (1974), 90–91.Google Scholar
Mañé, R.. Contributions to the stability conjecture. Topology 17 (1978), 383–396.Google Scholar
Mal’cev, A.. On a class of homogeneous spaces. Amer. Math. Soc. Transl. Ser. 2 39 (1951).Google Scholar
Manning, A.. There are no new Anosov diffeomorphisms on tori. Amer. J. Math 96 (1974), 422–442.CrossRefGoogle Scholar
Newhouse, S.. On codimension one Anosov diffeomorphisms. Amer. J. Math. 92 (1970), 761–770.Google Scholar
Novikov, S.. Topology of foliations (Russian). Tr. Mosk. Mat. Obs. 14 (1965), 248–277.Google Scholar
Osipenko, G., Ershov, E. and Kim, J. H.. Lectures on Invariant Manifolds of Perturbed Differential Equations and Linearization, St. Petersburg State Technical University, St. Petersburg, 1996. pp. 152.Google Scholar
Palmeira, C.. Open manifolds foliated by planes. Ann. of Math. (2) 107 (1978), 109–131.Google Scholar
Parwani, K.. On 3-manifolds that support partially hyperbolic diffeomorphisms. Nonlinearity 23 (2010), 589–606.Google Scholar
Peixoto, M.. Structural stability on two-dimensional manifolds. Topology 1 (1962), 101–120.Google Scholar
Plante, J. F.. Foliations of 3-manifolds with solvable fundamental group. Invent. Math. 51 (1979), 219–230.Google Scholar
Plante, J. F. and Thurston, W.. Anosov flows and the fundamental group. Topology 11 (1972), 147–150.Google Scholar
Potrie, R.. Partial hyperbolicity and foliations in T3 . J. Mod. Dyn. 9 (2015), 81–121.Google Scholar
Potrie, R.. Partial hyperbolic diffeomorphisms with a trapping property. Discrete Contin. Dyn. Syst. 35(10) (2015), 5037–5054.Google Scholar
Pujals, E. and Sambarino, M.. On the dynamics of dominated splittings. Ann. of Math. (2) 169 (2009), 675–740.Google Scholar
Pugh, C. and Shub, M.. Stable ergodicity. Bull. Amer. Math. Soc. 41 (2004), 1–41.Google Scholar
Hertz, M. A. R., Hertz, F. R. and Ures, R.. A survey on partially hyperbolic systems. Partially Hyperbolic Dynamics, Laminations and Teichmüller Flow (Fields Institute Communications, 51) . Eds. Forni, G., Lyubich, M., Pugh, C. and Shub, M.. American Mathematical Society, Providence, RI, 2007, pp. 35–87.Google Scholar
Hertz, M. A. R., Hertz, F. R. and Ures, R.. Tori with hyperbolic dynamics in 3-manifolds. J. Mod. Dyn. 51 (2011), 185–202.Google Scholar
Hertz, M. A. R., Hertz, F. R. and Ures, R.. A non dynamically coherent example in $\mathbb{T}^{3}$ . Ann. Inst. H. Poincaré Anal. Non Linéaire, to appear. Preprint 2014, arXiv:1409.0738.Google Scholar
Roberts, R., Shareshian, J. and Stein, M.. Infinitely many hyperbolic 3-manifolds which contain no Reebless foliations. J. Amer. Math. Soc. 16 (2003), 639–679.Google Scholar
Ruelle, D. and Sullivan, D.. Currents, flows and diffeomorphisms. Topology 14(4) (1975), 319–327.Google Scholar
Sadovskaya, V.. On uniformly quasiconformal Anosov systems. Math. Res. Lett. 12 (2005), 425–441.Google Scholar
Solodov, V. V.. Components of topological foliations. Math. USSR Sbornik 47 (1984), 329–343.Google Scholar
Sullivan, D.. A counterexample to the periodic orbit conjecture. Publ. Math. Inst. Hautes Études Sci. 46(1) (1976), 5–14.Google Scholar
Verjovsky, A.. Codimension one Anosov flows. Bol. Soc. Mat. Mexicana (3) 19(2) (1974), 49–77.Google Scholar
Wilkinson, A.. Conservative partially hyperbolic dynamics. Preprint, 2010, arXiv:1004.5345, Expository paper for the ICM 2010 Proceedings.Google Scholar