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Partially hyperbolic dynamics in dimension three

Published online by Cambridge University Press:  09 May 2017

PABLO D. CARRASCO ,
FEDERICO RODRIGUEZ-HERTZ ,
JANA RODRIGUEZ-HERTZ  and
RAÚL URES
PABLO D. CARRASCO
Affiliation:
ICMC-USP, Avenida Trabalhador São-carlense 400, São Carlos, SP 13566-590, Brazil email pdcarrasco@gmail.com
FEDERICO RODRIGUEZ-HERTZ
Affiliation:
PSU Mathematics Department, University Park, State College, PA 16802, USA email hertz@math.psu.edu
JANA RODRIGUEZ-HERTZ
Affiliation:
IMERL-FING, Julio Herrera y Reissig 565, Montevideo 11300, Uruguay email jana@fing.edu.uy, ures@fing.edu.uy
RAÚL URES
Affiliation:
IMERL-FING, Julio Herrera y Reissig 565, Montevideo 11300, Uruguay email jana@fing.edu.uy, ures@fing.edu.uy
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Abstract

Partial hyperbolicity appeared in the 1960s as a natural generalization of hyperbolicity. In the last 20 years, there has been great activity in this area. Here we survey the state of the art in some related topics, focusing especially on partial hyperbolicity in dimension three. The reason for this is not only that it is the smallest dimension in which non-degenerate partial hyperbolicity can occur, but also that the topology of $3$-manifolds influences the dynamics in revealing ways.

Information

Type
Survey Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 8 , December 2018 , pp. 2801 - 2837
Copyright
© Cambridge University Press, 2017 

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References

Avila, A., Crovisier, S. and Wilkinson, A.. Diffeomorphisms with positive metric entropy. Publ. Math. Inst. Hautes Études Sci. 124(1) (2016), 319–347.Google Scholar
Anosov, D.. Geodesic flows on Riemann manifolds with negative curvature. Proc. Steklov Inst. 90 (1967), 1–235.Google Scholar
Anosov, D. V. and Sinai., Y. G.. Certain smooth ergodic systems. Russian Math. Surveys 22 (1967), 103–167.Google Scholar
Barbot, T.. Generalizations of the Bonatti–Langevin example of Anosov flow and their classification up to topological equivalence. Comm. Anal. Geom. 6(4) (1998), 749–798.Google Scholar
Brin, M., Burago, D. and Ivanov, S.. On partially hyperbolic diffeomorphisms on 3-manifolds with commutative fundamental group. Advances in Dynamical Systems. 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.Google Scholar
Burns, K., Dolgopyat, D. and Pesin, Y.. Partial hyperbolicity, Lyapunov exponents and stable ergodicity. J. Stat. Phys. 108 (2002), 927–942. Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays.Google Scholar
Bonatti, C., Gogolev, A. and Potrie, R.. Anomalous partially hyperbolic diffeomorphisms II: stably ergodic examples. Invent. Math. 206(3) (2016), 801–836.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
Bonatti, C. and Langevin, R.. Un exemple de flot d’Anosov transitif transverse à un tore et non conjugué à une suspension. Ergod. Th. & Dynam. Sys. 14(4) (1994), 633–643.Google Scholar
Brin, M. and Pesin, Y.. Partially hyperbolic dynamical systems. Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 170–212.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:1411.1221v1.Google Scholar
Brin, M.. Seifert fibered spaces. Preprint, 1993, arXiv:0711.1346.Google Scholar
Brunella, M.. Expansive flows on seifert manifolds and on torus bundles. Bull. Braz. Math. Soc. (N.S.) 241 (1993), 89–104.Google Scholar
Bonatti, C. and Wilkinson, A.. Transitive partially hyperbolic diffeomorphisms on 3-manifolds. Topology 44(3) (2005), 475–508.Google Scholar
Burns, K. and Wilkinson, A.. Better center bunching. Preprint, 2005.Google Scholar
Burns, K. and Wilkinson, A.. Dynamical coherence and center bunching. Discrete Contin. Dyn. Syst. 22(1–2) (2008), 89–100.Google Scholar
Burns, K. and Wilkinson, A.. On the ergodicity of partially hyperbolic systems. Ann. of Math. (2) 171 (2010), 451–489.Google Scholar
Didier, P.. Stability of accessibility. Ergod. Th. & Dynam. Sys. 23(6) (2003), 1717–1731.Google Scholar
Dolgopyat, D. and Wilkinson, A.. Stable accessibility is C 1 dense. Astérisque 287 (2003.), 33–60.Google Scholar
Epstein, D. B. A.. Pointwise periodic homeomorphisms. Proc. Lond. Math. Soc. (3) s3‐42(3) (1981), 415–460.Google Scholar
Fenley, S.. Anosov flows in 3-manifolds. Ann. of Math. (2) 139(1) (1994), 79–115.Google Scholar
Fried, D.. Transitive anosov flows and pseudo-anosov maps. Topology 22(3) (1983), 299–303.Google Scholar
Franks, J. and Williams, R.. Anomalous Anosov flows. Global Theory of Dynamical Systems (Proc. Int. Conf., Northwestern University, Evanston, IL, 1979 (Lecture Notes in Mathematics, 819) . Springer, Berlin, 1980, pp. 158–174.Google Scholar
Franks, J. and Williams, B.. Anomalous Anosov flows. Global Theory of Dynamical Systems (Proc. Int. Conf., Northwestern University, Evanston, IL, 1979 (Lecture Notes in Mathematics, 819) . Springer, Berlin, 1980, pp. 158–174.Google Scholar
Grayson, M., Pugh, C. and Shub, M.. Stably ergodic diffeomorphisms. Ann. of Math. (2) 140 (1994), 295–329.Google Scholar
Haefliger, A.. Variétés feuilletées. Topologia Differenziale (Centro Internaz. Mat. Estivo, 1deg Ciclo, Urbino, 1962), Lezione 2. Edizioni Cremonese, Rome, 1962, p. 46.Google Scholar
Hammerlindl, A.. Leaf conjugacies on the torus. Ergod. Th. & Dynam. Sys. 3(33) (2013), 896–933.Google Scholar
Hatcher, A.. Notes on basic 3-manifold topology. Available at: www.math.cornell.edu/ hatcher/, 2007.Google Scholar
Rodriguez Hertz, F., Rodriguez Hertz, M. A., Tahzibi, A. and Ures, R.. New criteria for ergodicity and nonuniform hyperbolicity. Duke Math. J. 160(3) (2011), 599–629.Google Scholar
Rodriguez Hertz, F., Rodriguez Hertz, M. A. and Ures, R.. Partial hyperbolicity and ergodicity in dimension three. J. Mod. Dyn. 2 (2008), 187–208.Google Scholar
Rodriguez Hertz, F., Rodriguez Hertz, M. and Ures, R.. Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1d-center bundle. Invent. Math. 2(172) (2008), 353–381.Google Scholar
Rodriguez Hertz, F., Rodriguez Hertz, M. and Ures, R.. On the existence and uniqueness of weak foliations in dimension 3. Geom. Probab. Struct. Dyn. 469 (2008), 303–316.Google Scholar
Rodriguez Hertz, F., Rodriguez Hertz, M. and Ures, R.. Tori with hyperbolic dynamics in 3-manifolds. J. Mod. Dyn. 5(1) (2011), 185–202.Google Scholar
Rodriguez Hertz, F., Rodriguez Hertz, M. and Ures, R.. Center-unstable foliations do not have compact leaves. Math. Res. Lett. (2015), to appear.Google Scholar
Rodriguez Hertz, F., Rodriguez Hertz, M. A. and Ures, R.. A non-dynamically coherent example on T3 . Ann. Inst. Henri-Poincaré 33(4) (2016), 1023–1032.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
Hammerlindl, A. and Potrie, R.. Pointwise partial hyperbolicity in 3-dimensional nilmanifolds. J. Lond. Math. Soc. (2) 3(89) (2014), 853–875.Google Scholar
Hammerlindl, A. and Potrie, R.. Partial hyperbolicity and classification: a survey. Ergod. Th. & Dynam. Sys., to appear, Preprint, 2015, arXiv:1511.04471.Google Scholar
Hirsch, M., Pugh, C. and Shub, M.. Invariant Manifolds. Springer, Berlin, 1977.Google Scholar
Handel, M. and Thurston, W. P.. Anosov flows on new three manifolds. Invent. Math. 59(2) (1980), 95–103.Google Scholar
Journé, J.-L.. A regularity lemma for functions of several variables. Rev. Mat. Iberoam. 4 (1988), 187–193.Google Scholar
Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54) . Cambridge University Press, Cambridge, 1995.Google Scholar
Mañé, R.. Contributions to the stability conjecture. Topology 17 (1978), 383–396.Google Scholar
Moreira, C. G. and Silva, W.. On the geometry of horseshoes in higher dimensions. Preprint, 2012, arXiv:1210.2623.Google Scholar
Potrie, R.. Partial hyperbolicity and foliations in t3 . J. Mod. Dyn. 9(1) (2015), 81–121.Google Scholar
Pugh, C. and Shub, M.. Ergodicity of anosov actions. Invent. Math. 15 (1972), 1–23.Google Scholar
Pugh, C. and Shub, M.. Stable ergodicity and partial hyperbolicity. Proc. 1st Int. Conf. on Dynamical Systems (Montevideo, Uruguay, 1995) (Research Notes Mathematics Series, 362) . Eds. Newhouse, S., Ledrappier, F. and Lewowicz, J.. Longman, Pitman, Harlow, 1996, pp. 182–187.Google Scholar
Pugh, C. and Shub., M.. Stably ergodic dynamical systems and partial hyperbolicity. J. Complexity 13 (1997), 125–179.Google Scholar
Pugh, C. and Shub, M.. Stable ergodicity and julienne quasiconformality. J. Eur. Math. Soc. (JEMS) 2(1) (2000), 1–52.Google Scholar
Pugh, C., Shub, M. and Wilkinson, A.. Holder foliations. Duke Math. J. 86(3) (1997), 517–546.Google Scholar
Rosenberg, H.. Foliations by planes. Topology 7(2) (1968), 131–138.Google Scholar
Roussarie, R.. Sur les feuilletages des variétés de dimension trois. Ann. Inst. Fourier (Grenoble) 21 (1971), 13–82.Google Scholar
Sacksteder, R.. Strongly mixing transformations. Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, CA, 1968). American. Mathematical Society, Providence, RI, 1970, pp. 245–252.Google Scholar
Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. (N.S.) 73 (1967), 747–817.Google Scholar
Wilkinson, A.. Stable ergodicity of the time-one map of a geodesic flow. Ergod. Th. & Dynam. Sys. 18(6) (1998), 1545–1587.Google Scholar