Hostname: page-component-669899f699-8p65j Total loading time: 0 Render date: 2025-04-25T12:32:06.241Z Has data issue: false hasContentIssue false
  • English
  • Français

Smooth models for certain fibered partially hyperbolic systems

Part of: Dynamical systems with hyperbolic behavior Smooth dynamical systems: general theory

Published online by Cambridge University Press:  13 November 2023

MEG DOUCETTE
MEG DOUCETTE*
Affiliation:
Department of Mathematics, University of Chicago, Chicago, USA
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that under restrictions on the fiber, any fibered partially hyperbolic system over a nilmanifold is leaf conjugate to a smooth model that is isometric on the fibers and descends to a hyperbolic nilmanifold automorphism on the base. One ingredient is a result of independent interest generalizing a result of Hiraide: an Anosov homeomorphism of a nilmanifold is topologically conjugate to a hyperbolic nilmanifold automorphism.

Keywords

partial hyperbolicitysmooth modelsrigidityAnosov homeomorphisms

MSC classification

Primary: 37D30: Partially hyperbolic systems and dominated splittings
Secondary: 37C15: Topological and differentiable equivalence, conjugacy, invariants, moduli, classification
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 8 , August 2024 , pp. 2173 - 2198
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press

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.)

Article purchase

Temporarily unavailable

References

Anderson, R. D. and Bing, R. H.. A complete elementary proof that Hilbert space is homeomorphic to the countable infinite product of lines. Bull. Amer. Math. Soc. (N.S.) 74(5) (1968), 771–792.10.1090/S0002-9904-1968-12044-0CrossRefGoogle Scholar
Aoki, N. (1983). On homeomorphisms with pseudo-orbit tracing property. Tokyo J. Math. 06(2), 329–334.10.3836/tjm/1270213874CrossRefGoogle Scholar
Avila, A., Viana, M. and Wilkinson, A.. Absolute continuity, Lyapunov exponents, and rigidity II: systems with compact center leaves. Ergod. Th. & Dynam. Sys. 42(2) (2022), 437–490.10.1017/etds.2021.42CrossRefGoogle Scholar
Bamler, R. and Kleiner, B.. Ricci flow and diffeomorphism groups of 3-manifolds. J. Amer. Math. Soc. 36(2) (2022), 563–589.10.1090/jams/1003CrossRefGoogle Scholar
Bohnet, D.. Codimension one partially hyperbolic diffeomorphisms with a uniformly compact center foliation. J. Mod. Dyn. 7(4) (2013), 565–604.10.3934/jmd.2013.7.565CrossRefGoogle Scholar
Bohnet, D. and Bonatti, C.. Partially hyperbolic diffeomorphisms with a uniformly compact center foliation: the quotient dynamics. Ergod. Th. & Dynam. Sys. 36(4) (2016), 1067–1105.10.1017/etds.2014.102CrossRefGoogle Scholar
Candel, A. and Conlon, L.. Foliations I (Graduate Studies in Mathematics, 23). American Mathematical Society, Providence, RI, 1999.10.1090/gsm/023CrossRefGoogle Scholar
Cerf, J.. La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie. Publ. Math. Inst. Hautes Études Sci. 39 (1970), 5–173.10.1007/BF02684687CrossRefGoogle Scholar
Crowley, D. and Schick, T.. The Gromoll filtration, KO–characteristic classes and metrics of positive scalar curvature. Geom. Topol. 17(3) (2013), 1773–1789.10.2140/gt.2013.17.1773CrossRefGoogle Scholar
Damjanović, D., Wilkinson, A. and Xu, D.. Pathology and asymmetry: centralizer rigidity for partially hyperbolic diffeomorphisms. Duke Math. J. 170(17) (2021), 3815–3890.10.1215/00127094-2021-0053CrossRefGoogle Scholar
Dold, A.. Fixed point index and fixed point theorem for Euclidean neighborhood retracts. Topology 4(1) (1965), 1–8.10.1016/0040-9383(65)90044-3CrossRefGoogle Scholar
Farb, B. and Margalit, D.. A Primer on Mapping Class Groups (Princeton Mathematical Series, 49). Princeton University Press, Princeton, NJ, 2011.10.1515/9781400839049CrossRefGoogle Scholar
Franks, J.. Anosov diffeomorphisms on tori. Trans. Amer. Math. Soc. 145 (1969), 117–124.10.1090/S0002-9947-1969-0253352-7CrossRefGoogle Scholar
Franks, J.. Anosov Diffeomorphims. Global Analysis (Proceedings of Symposia in Pure Mathematics, 14). Eds. Chern, S.-S. and Smale, S.. American Mathematical Society, Providence, RI, 1970, pp. 61–93.10.1090/pspum/014/0271990CrossRefGoogle Scholar
Gabai, D.. The Smale conjecture for hyperbolic 3-manifolds: Isom(m3) $\simeq$ Diff(m3). J. Differential Geom. 58 (2001), 113–149.10.4310/jdg/1090348284CrossRefGoogle Scholar
Gogolev, A., Ontaneda, P. and Rodriguez Hertz, F.. New partially hyperbolic dynamical systems I. Acta Math. 215(2) (2015), 363–393.10.1007/s11511-016-0135-3CrossRefGoogle Scholar
Gorbatsevich, V. V., Onishchik, A. L. and Vinberg, E. B.. Lie Groups and Lie Algebras I: Foundations of Lie Theory Lie Transformation Groups (Encyclopaedia of Mathematical Sciences, Lie Groups and Lie Algebras, 20). Springer-Verlag, Berlin, 1993.Google Scholar
Hammerlindl, A.. Leaf conjugacies on the torus. Ergod. Th. & Dynam. Sys. 33(3) (2013), 896–933.10.1017/etds.2012.171CrossRefGoogle Scholar
Hammerlindl, A. and Potrie, R.. Pointwise partial hyperbolicity in three-dimensional nilmanifolds. J. Lond. Math. Soc. (2) 89 (2013), 853–875.10.1112/jlms/jdu013CrossRefGoogle Scholar
Hatcher, A.. Homeomorphisms of sufficiently large ${p}^2$ -irreducible 3-manifolds. Topology 15(4) (1976), 343–347.10.1016/0040-9383(76)90027-6CrossRefGoogle Scholar
Hatcher, A.. A proof of the Smale conjecture, Diff( ${s}^3$ ) $\simeq$ O(4). Ann. of Math. (2) 117(3) (1983), 553–607.10.2307/2007035CrossRefGoogle Scholar
Hatcher, A.. A 50-year view of diffeomorphism groups. Notes. Available at https://pi.math.cornell.edu/~hatcher/Papers/Diff%28M%292012.pdf (2012).Google Scholar
Hatcher, A.. Spaces of incompressible surfaces. Available at https://pi.math.cornell.edu/~hatcher/Papers/emb.pdf (no date). (This is an updated version of [20])Google Scholar
Hiraide, K.. Expansive homeomorphisms with the pseudo-orbit tracing property of n-tori. J. Math. Soc. Japan 41(3) (1989), 357–389.10.2969/jmsj/04130357CrossRefGoogle Scholar
Hirsch, M. W., Pugh, C. C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583). Springer-Verlag, Berlin, 1977.10.1007/BFb0092042CrossRefGoogle Scholar
Hong, S., Kalliongis, J., McCullough, D. and Rubinstein, J. H.. Diffeomorphisms of Elliptic 3-Manifolds (Lecture Notes in Mathematics, 2055). Springer, Berlin, 2012.10.1007/978-3-642-31564-0CrossRefGoogle Scholar
Husemöller, D.. Fibre Bundles (Graduate Texts in Mathematics, 20), 3rd edn. Springer-Verlag, New York, 1994.10.1007/978-1-4757-2261-1CrossRefGoogle Scholar
Lee, J.. Introduction to Smooth Manifolds (Graduate Texts in Mathematics, 218), 2nd edn. Springer-Verlag, New York, 2012.10.1007/978-1-4419-9982-5CrossRefGoogle Scholar
Mane, R.. Ergodic Theory and Differentiable Dynamics (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/A Series of Modern Surveys in Mathematics, 8). Springer-Verlag, Berlin, 1987.10.1007/978-3-642-70335-5CrossRefGoogle Scholar
Manning, A.. Anosov diffeomorphisms on nilmanifolds. Proc. Amer. Math. Soc. 38(2) (1973), 423–426.10.1090/S0002-9939-1973-0343317-5CrossRefGoogle Scholar
Manning, A.. There are no new Anosov diffeomorphisms on tori. Amer. J. Math. 96(3) (1974), 422–429.10.2307/2373551CrossRefGoogle Scholar
Milnor, J.. On manifolds homeomorphic to the 7-sphere. Ann. of Math. (2) 64(2) (1956), 399–405.10.2307/1969983CrossRefGoogle Scholar
Morita, S.. Geometry of Characteristic Classes (Translations of Mathematical Monographs, 199). American Mathematical Society, Providence, RI, 2001.10.1090/mmono/199CrossRefGoogle Scholar
Müller, C. and Wockel, C.. Equivalences of smooth and continuous principal bundles with infinite-dimensional structure group. Adv. Geom. 9(4) (2009), 605–626.10.1515/ADVGEOM.2009.032CrossRefGoogle Scholar
Myers, S. B. and Steenrod, N. E.. The group of isometries of a Riemannian manifold. Ann. of Math. (2) 40(2) (1939), 400–416.10.2307/1968928CrossRefGoogle Scholar
Niţică, V. and Török, A.. Local rigidity of certain partially hyperbolic actions of product type. Ergod. Th. & Dynam. Sys. 21(4) (2001), 1213–1237.10.1017/S0143385701001560CrossRefGoogle Scholar
Raghunathan, M. S.. Discrete Subgroups of Lie Groups (Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge, 68). Springer-Verlag, Berlin, 1972.10.1007/978-3-642-86426-1CrossRefGoogle Scholar
Smale, S.. Diffeomorphisms of the 2-Sphere. Proc. Amer. Math. Soc., 10(4) (1959), 621–626.10.1090/S0002-9939-1959-0112149-8CrossRefGoogle Scholar
Smale, S.. Generalized Poincare’s conjecture in dimensions greater than four. Ann. of Math. (2) 74(2) (1961), 391–406.10.2307/1970239CrossRefGoogle Scholar
Smrekar, J. and Yamashita, A.. Function spaces of CW homotopy type are Hilbert manifolds. Proc. Amer. Math. Soc. 137(2) (2008), 751–759.10.1090/S0002-9939-08-09584-1CrossRefGoogle Scholar
Sumi, N.. Topological Anosov maps of infra-nil-manifolds. J. Math. Soc. Japan 48(4) (1996), 607–648.10.2969/jmsj/04840607CrossRefGoogle Scholar
Waldhausen, F.. On irreducible 3-manifolds which are sufficiently large. Ann. of Math. (2) 87(1) (1968), 56–88.10.2307/1970594CrossRefGoogle Scholar
Watanabe, T.. Some exotic nontrivial elements of the rational homotopy groups of Diff( ${s}^4$ ). Preprint, 2019, arXiv:1812.02448.Google Scholar