Skip to main content Accessibility help
×
×
Home

Partial hyperbolicity and classification: a survey

  • ANDY HAMMERLINDL (a1) and RAFAEL POTRIE (a2)

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.

Copyright

References

Hide All
[An] Anosov, D.. Geodesic flows on closed Riemannian manifolds of negative curvature. Trudy Mat. Inst. Steklov. 90 (1967), 3210.
[ArP] Araújo, V. and Pacifico, M. J.. Three Dimensional Flows (Ergebnisse der Mathematik und ihrer Grenzgebiete) . Springer, Berlin, 2010.
[Ba] Barbot, T.. Flots d’Anosov sur les variétés graphées au sens de Waldhausen. Ann. Inst. Fourier (Grenoble) 46 (1996), 14511517.
[BaF] Barbot, T. and Fenley, S.. Pseudo Anosov flows in toroidal manifolds. Geom. Topol. 17 (2013), 18771954.
[BBY] Beguin, F., Bonatti, C. and Yu, B.. Building Anosov flows on 3-manifolds. Preprint, 2014, arXiv:1408.3951.
[BDP] 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), 355418.
[BoD] Bonatti, C. and Díaz, L.. Persistent transitive diffeomorphisms. Ann. of Math. (2) 143(2) (1996), 357396.
[BoDV] 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.
[BoGHP] Bonatti, C., Gogolev, A., Hammerlindl, A. and Potrie, R.. Anomalous partially hyperbolic diffeomorphisms III: Pseudo Anosov mapping classes. In preparation.
[BoGP] 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.
[Boh] Bohnet, D.. Codimension-1 partially hyperbolic diffeomorphisms with a uniformly compact center foliation. J. Mod. Dyn. 7 (2013), 565604.
[BoL] 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), 633643.
[BoPP] 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.
[BoW] Bonatti, C. and Wilkinson, A.. Transitive partially hyperbolic diffeomorphisms on 3-manifolds. Topology 44 (2005), 475508.
[BoZ] 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.
[BP] Brin, M. and Pesin, Y.. Partially hyperbolic dynamical systems. Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 170212.
[Br1] Brin, M.. Topological transitivity of one class of dynamic systems and flows of frames on manifolds of negative curvature. Funct. Anal. Appl. 9 (1975), 816.
[Br2] Brin, M.. On dynamical coherence. Ergod. Th. & Dynam. Sys. 23 (2003), 395401.
[BrBI1] 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. 307312.
[BrBI2] Brin, M., Burago, D. and Ivanov, S.. Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus. J. Mod. Dyn. 3 (2009), 111.
[BrG] Brin, M. and Gromov, M.. On the ergodicity of frame flows. Invent. Math. 60 (1980), 18.
[Bri] Brittenham, M.. Essential laminations in Seifert fibered spaces. Topology 32 (1993), 6185.
[BrM] 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. 4853.
[BuI] Burago, D. and Ivanov, S.. Partially hyperbolic diffeomorphisms of 3-manifolds with abelian fundamental groups. J. Mod. Dyn. 2 (2008), 541580.
[BuPSW1] Burns, K., Pugh, C., Shub, M. and Wilkinson, A.. Recent results about stable ergodicity. Proc. Sympos. Amer. Math. Soc. 69 (2001), 327366.
[BuW] Burns, K. and Wilkinson, A.. Dynamical coherence and center bunching. Discrete Contin. Dyn. Syst. 22 (2008), 89100.
[Cal] Calegari, D.. Foliations and the Geometry of 3-manifolds. Clarendon Press, Oxford, 2008.
[CalD] Calegari, D. and Dunfield, N.. Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), 149204.
[Carr] Carrasco, P.. Compact dynamical foliations. Ergod. Th. & Dynam. Syst. 35(8) (2015), 24742498.
[CC] Candel, A. and Conlon, L.. Foliations I and II (Graduate studies in Mathematics, 60) . American Mathematical Society, Providence, RI, 2003.
[CP] 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.
[CRHRHU] Carrasco, P., Hertz, M. A. R., Hertz, F. R. and Ures, R.. Partially hyperbolic dynamics in dimension 3. Preprint, 2015, arXiv:1501.00932.
[DoP] Dolgopyat, D. and Pesin, Y.. Every compact manifold carries a completely hyperbolic diffeomorphism. Ergod. Th. & Dynam. Sys. 22 (2002), 409437.
[DPU] Díaz, L., Pujals, E. and Ures, R.. Partial hyperbolicity and robust transitivity. Acta Math. 183 (1999), 143.
[Fen] Fenley, S.. Anosov flows in 3-manifolds. Ann. of Math. (2) 139(1) (1994), 79115.
[FG1] Farrell, F. T. and Gogolev, A.. Anosov diffeomorphisms constructed from 𝜋 k (Diff(S n )). J. Topol. 5 (2012), 276292.
[FG2] Farrell, F. T. and Gogolev, A.. On bundles that admit fiberwise hyperbolic dynamics. Math. Ann. 364(1) (2016), 401438.
[FJ] Farrell, T. F. and Jones, L.. Anosov diffeomorphisms constructed from 𝜋1(Diff(S n )). Topology 17 (1978), 273282.
[FPS] Fisher, T., Potrie, R. and Sambarino, M.. Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to Anosov. Math. Z. 278 (2014), 149168.
[Fr] 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. 6193.
[FrW] 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. 158174.
[Fu] Fuller, F. B.. On the surface of section and periodic trajectories. Amer. J. Math. 87 (1965), 473480.
[Ghy] Ghys, E.. Flots d’Anosov sur les 3-variétés fibrées en cercles. Ergod. Th. & Dynam. Sys. 4 (1984), 6780.
[Go] Gogolev, A.. Partially hyperbolic diffeomorphisms with compact center foliations. J. Mod. Dyn. 5 (2011), 747767.
[GoL] 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.
[Goo] Goodman, S.. Dehn surgery on Anosov flows. Geometric Dynamics (Rio de Janeiro, 1981) (Lecture Notes in Mathematics, 1007) . Springer, Berlin, 1983, pp. 300307.
[GoORH] Gogolev, A., Ontaneda, P. and Hertz, F. R.. New partially hyperbolic dynamical systems I. Preprint, 2014, arXiv:1407.7768.
[GoRH] Gogolev, A. and Hertz, F. R.. Manifolds with higher homotopy which do not support Anosov diffeomorphisms. Bull. Lond. Math. Soc. 46(2) (2014), 349366.
[GP] Gourmelon, N. and Potrie, R.. Projectively Anosov diffeomorphisms of surfaces. In preparation.
[GPS] Grayson, M., Pugh, C. and Shub, M.. Stably ergodic diffeomorphisms. Ann. of Math. (2) 140(2) (1994), 295330.
[Gr1] Gromov, M.. Three remarks on geodesic dynamics and fundamental group. Preprint SUNY, 1976. Enseign. Math. 46 (2000), 391402 reprinted.
[Gr2] Gromov, M.. Groups of polynomial growth and expanding maps. Publ. Math. Inst. Hautes Études Sci. 53 (1981), 5373.
[Ha1] Hammerlindl, A.. Leaf conjugacies in the torus. Ergod. Th. & Dynam. Sys. 33(3) (2013), 896933.
[Ha2] Hammerlindl, A.. Integrability and Lyapunov exponents. J. Mod. Dyn. 5(1) (2011), 107122.
[Ha3] Hammerlindl, A.. Dynamics of quasi-isometric foliations. Nonlinearity 25 (2012), 15851599.
[Ha4] Hammerlindl, A.. Partial hyperbolicity on 3-dimensional nilmanifolds. Discrete Contin. Dyn. Syst. 33(8) (2013), 36413669.
[Ha5] Hammerlindl, A.. Polynomial global product structure. Proc. Amer. Math. Soc. 142(12) (2014), 42974303.
[Ha6] Hammerlindl, A.. On expanding foliations. Bull. Braz. Math. Soc. (N.S.) 46(3) (2015), 407420.
[Ha7] Hammerlindl, A.. Ergodic components of partially hyperbolic systems. Preprint, 2014,arXiv:1409.8002.
[HaP1] Hammerlindl, A. and Potrie, R.. Pointwise partial hyperbolicity in 3-dimensional nilmanifolds. J. Lond. Math. Soc. 89(3) (2014), 853875.
[HaP2] Hammerlindl, A. and Potrie, R.. Classification of partially hyperbolic diffeomorphisms in 3-manifolds with solvable fundamental group. J. Topol. 8(3) (2015), 842870.
[HaPe] 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. 155.
[HaPS] Hammerlindl, A., Potrie, R. and Shannon, M.. Seifert manifolds admitting partially hyperbolic diffeomorphisms. In preparation.
[Hat] Hatcher, A.. Algebraic Topology. Cambridge University Press, Cambridge, 2002.
[HaTh] Handel, M. and Thurston, W.. Anosov flows on new three manifolds. Invent. Math. 59(2) (1980), 95103.
[HaU] Hammerlindl, A. and Ures, R.. Ergodicity and partial hyperbolicity on the 3-torus. Commun. Contemp. Math. 16(4) (2014), 135158.
[Hi] Hirsch, M.. Differential Topology (Springer Graduate Texts in Mathematics, 33) . Springer, New York, 1976.
[HPS] Hirsch, M., Pugh, C. and Shub, M.. Invariant Manifolds (Springer Lecture Notes in Mathematics, 583) . Springer, Berlin, 1977.
[Hun] Hungerford, T.. Algebra (Springer Graduate Texts in Mathematics, 73) . Reprint of the 1974 original. Springer, Berlin, 1980.
[Le] Levitt, G.. Feuilletages des variétés de dimension 3 qui sont fibrés en circles. Comment. Math. Helv. 53 (1978), 572594.
[LTW] Luzzatto, S., Turelli, S. and War, K.. Integrability of dominated decompositions on three-dimensional manifolds. Preprint, 2014, arXiv:1410.8072.
[Ma1] Mañé, R.. Persistent manifolds are normally hyperbolic. Bull. Amer. Math. Soc. 80(1) (1974), 9091.
[Ma2] Mañé, R.. Contributions to the stability conjecture. Topology 17 (1978), 383396.
[Mal] Mal’cev, A.. On a class of homogeneous spaces. Amer. Math. Soc. Transl. Ser. 2 39 (1951).
[Man] Manning, A.. There are no new Anosov diffeomorphisms on tori. Amer. J. Math 96 (1974), 422442.
[Ne] Newhouse, S.. On codimension one Anosov diffeomorphisms. Amer. J. Math. 92 (1970), 761770.
[No] Novikov, S.. Topology of foliations (Russian). Tr. Mosk. Mat. Obs. 14 (1965), 248277.
[OEK] 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.
[Pal] Palmeira, C.. Open manifolds foliated by planes. Ann. of Math. (2) 107 (1978), 109131.
[Par] Parwani, K.. On 3-manifolds that support partially hyperbolic diffeomorphisms. Nonlinearity 23 (2010), 589606.
[Pei] Peixoto, M.. Structural stability on two-dimensional manifolds. Topology 1 (1962), 101120.
[Pl] Plante, J. F.. Foliations of 3-manifolds with solvable fundamental group. Invent. Math. 51 (1979), 219230.
[PlT] Plante, J. F. and Thurston, W.. Anosov flows and the fundamental group. Topology 11 (1972), 147150.
[Pot1] Potrie, R.. Partial hyperbolicity and foliations in T3 . J. Mod. Dyn. 9 (2015), 81121.
[Pot2] Potrie, R.. Partial hyperbolic diffeomorphisms with a trapping property. Discrete Contin. Dyn. Syst. 35(10) (2015), 50375054.
[PujS] Pujals, E. and Sambarino, M.. On the dynamics of dominated splittings. Ann. of Math. (2) 169 (2009), 675740.
[PuSh] Pugh, C. and Shub, M.. Stable ergodicity. Bull. Amer. Math. Soc. 41 (2004), 141.
[RHRHU1] 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. 3587.
[RHRHU2] Hertz, M. A. R., Hertz, F. R. and Ures, R.. Tori with hyperbolic dynamics in 3-manifolds. J. Mod. Dyn. 51 (2011), 185202.
[RHRHU3] 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.
[RSS] Roberts, R., Shareshian, J. and Stein, M.. Infinitely many hyperbolic 3-manifolds which contain no Reebless foliations. J. Amer. Math. Soc. 16 (2003), 639679.
[RuS] Ruelle, D. and Sullivan, D.. Currents, flows and diffeomorphisms. Topology 14(4) (1975), 319327.
[Sa] Sadovskaya, V.. On uniformly quasiconformal Anosov systems. Math. Res. Lett. 12 (2005), 425441.
[So] Solodov, V. V.. Components of topological foliations. Math. USSR Sbornik 47 (1984), 329343.
[Su] Sullivan, D.. A counterexample to the periodic orbit conjecture. Publ. Math. Inst. Hautes Études Sci. 46(1) (1976), 514.
[Ve] Verjovsky, A.. Codimension one Anosov flows. Bol. Soc. Mat. Mexicana (3) 19(2) (1974), 4977.
[Wi] Wilkinson, A.. Conservative partially hyperbolic dynamics. Preprint, 2010, arXiv:1004.5345, Expository paper for the ICM 2010 Proceedings.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed