Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T21:45:53.919Z Has data issue: false hasContentIssue false
  • English
  • Français

Centralizers of derived-from-Anosov systems on ${\mathbb T}^3$: rigidity versus triviality

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

Published online by Cambridge University Press:  02 August 2021

SHAOBO GAN ,
YI SHI ,
DISHENG XU  and
JINHUA ZHANG
SHAOBO GAN
Affiliation:
School of Mathematical Sciences, Peking University, Beijing 100871, P. R. China (e-mail: gansb@pku.edu.cn, shiyi@math.pku.edu.cn)
YI SHI
Affiliation:
School of Mathematical Sciences, Peking University, Beijing 100871, P. R. China (e-mail: gansb@pku.edu.cn, shiyi@math.pku.edu.cn)
DISHENG XU
Affiliation:
Beijing International Center for Mathematical Research, Peking University, Beijing 100871, P. R. China (e-mail: xudisheng@bicmr.pku.edu.cn)
JINHUA ZHANG*
Affiliation:
School of Mathematical Sciences, Beihang University, Beijing 100191, P. R. China (e-mail: jinhua_zhang@buaa.edu.cn)
*
e-mail: zjh200889@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we study the centralizer of a partially hyperbolic diffeomorphism on ${\mathbb T}^3$ which is homotopic to an Anosov automorphism, and we show that either its centralizer is virtually trivial or such diffeomorphism is smoothly conjugate to its linear part.

Keywords

centralizerpartial hyperbolicityAnosov diffeomorphismrigidity

MSC classification

Primary: 37D30: Partially hyperbolic systems and dominated splittings
Secondary: 37C05: Smooth mappings and diffeomorphisms 37C15: Topological and differentiable equivalence, conjugacy, invariants, moduli, classification 37C85: Dynamics of group actions other than ${bf Z}$ and ${bf R}$, and foliations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 9 , September 2022 , pp. 2841 - 2865
Check for updates
Copyright
© The Author(s), 2021. 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.)

References

Adler, R. L. and Palais, R.. Homeomorphic conjugacy of automorphisms on the torus. Proc. Amer. Math. Soc. 16 (1965), 1222–1225.CrossRefGoogle Scholar
Bonatti, C.. Un point fixe commun pour des difféomorphismes commutants de ${S}^2$ . Ann. of Math. (2) 129(1) (1989), 61–69.CrossRefGoogle Scholar
Brin, M., Burago, D. and Ivanov, S.. Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus. J. Mod. Dyn. 3(1) (2009), 1–11.CrossRefGoogle Scholar
Brin, M., Burago, D. and Ivanov, S.. On partially hyperbolic diffeomorphisms of 3-manifolds with commutative fundamental group. Modern Dynamical Systems and Applications. Ed. M. Brin, B. Hasselblatt and Y. Pesin. Cambridge University Press, Cambridge, 2004, pp. 307–312.Google Scholar
Bonatti, C., Crovisier, S., Vago, G. and Wilkinson, A.. Local density of diffeomorphisms with large centralizers. Ann. Sci. Éc. Norm. Supér. (4) 41(6) (2008), 925–954.CrossRefGoogle Scholar
Bonatti, C., Crovisier, S. and Wilkinson, A.. The ${C}^1$ generic diffeomorphism has trivial centralizer. Publ. Math. Inst. Hautes Études Sci. 109 (2009), 185–244.CrossRefGoogle Scholar
Buzzi, J., Fisher, T., Sambarino, M. and Vásquez, C.. Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems. Ergod. Th. & Dynam. Sys. 32(1) (2012), 63–79.CrossRefGoogle Scholar
Barthelmé, T. and Gogolev, A.. Centralizers of partially hyperbolic diffeomorphisms in dimension 3. Discrete Contin. Dyn. Syst. 41(9) (2021), 4477–4484.CrossRefGoogle Scholar
Burago, D. and Ivanov, S.. Partially hyperbolic diffeomorphisms of 3-manifolds with abelian fundamental groups. J. Mod. Dyn. 2(4) (2008), 541–580.CrossRefGoogle Scholar
Brin, M.. On dynamical coherence. Ergod. Th. & Dynam. Sys. 23(2) (2003), 395–401.CrossRefGoogle Scholar
Burslem, L.. Centralizers of partially hyperbolic diffeomorphisms. Ergod. Th. & Dynam. Sys. 24(1) (2004), 55–87.CrossRefGoogle Scholar
Didier, P.. Stability of accessibility. Ergod. Th. & Dynam. Sys. 23(6) (2003), 1717–1731.CrossRefGoogle Scholar
Damjanovic, D. and Katok, A.. Local rigidity of partially hyperbolic actions I. KAM method and ${\mathbb{Z}}^k$ actions on the torus. Ann. of Math. (2) 172(3) (2010), 1805–1858.CrossRefGoogle Scholar
Dolgopyat, D. and Wilkinson, A.. Stable accessibility is ${C}^1$ dense. Geometric Methods in Dynamics. II (Astérisque, 287). Eds. W. de Melo, M. Viana and J. C. Yoccoz. Société Mathématique de France, Paris, 2003, pp. xvii, 33–60.Google Scholar
Damjanovic, D., Wilkinson, A. and Xu, D.. Pathology and asymmetry: centralizer rigidity for partially hyperbolic diffeomorphisms. Duke Math J. to appear.Google Scholar
Fenley, S.. Quasi-isometric foliations. Topology 31(3) (1992), 667–676.CrossRefGoogle Scholar
Fisher, T.. Trivial centralizers for Axiom A diffeomorphisms. Nonlinearity 21(11) (2008), 2505–2517.CrossRefGoogle Scholar
Fisher, T., Potrie, R. and Sambarino, M.. Dynamical coherence of partially hyperbolic diffeomorphisms of tori homotopic to Anosov. Math. Z. 278(1–2) (2014), 149–168.CrossRefGoogle Scholar
Franks, J.. Anosov diffeomorphisms. Global Analysis (Proceedings of Symposia in Pure Mathematics, 14). Eds. S. S. Chern and S. Smale. American Mathematical Society, Providence, RI, 1970, pp. 61–93.Google Scholar
Gogolev, A.. How typical are pathological foliations in partially hyperbolic dynamics: an example. Israel J. Math. 187 (2012), 493–507.CrossRefGoogle Scholar
Gogolev, A.. Bootstrap for local rigidity of Anosov automorphisms on the 3-torus. Comm. Math. Phys. 352(2) (2017), 439–455.CrossRefGoogle Scholar
Ren, Y., Gan, S. and Zhang, P.. Accessibility and homology bounded strong unstable foliation for Anosov diffeomorphisms on 3-torus. Acta Math. Sin. (Engl. Ser.) 33(1) (2017), 71–76.CrossRefGoogle Scholar
Gan, S. and Shi, Y.. Rigidity of center Lyapunov exponents and su-integrability. Comment. Math. Helv. 95(3) (2020), 569–592.CrossRefGoogle Scholar
Rodriguez Hertz, F.. Global rigidity of certain abelian actions by toral automorphisms. J. Mod. Dyn. 1(3) (2007), 425–442.CrossRefGoogle Scholar
Hammerlindl, A.. Leaf conjugacies on the torus. Ergod. Th. & Dynam. Sys. 33(3) (2013), 896–933.CrossRefGoogle Scholar
Hammerlindl, A. and Potrie, R.. Pointwise partial hyperbolicity in three-dimensional nilmanifolds. J. Lond. Math. Soc. (2) 89(3) (2014), 853–875.CrossRefGoogle Scholar
Hammerlindl, A. and Shi, Y.. Accessibility of derived-from-Anosov systems. Trans. Amer. Math. Soc. 374(4) (2021), 2949–2966.CrossRefGoogle Scholar
Hammerlindl, A. and Ures, R.. Ergodicity and partial hyperbolicity on the 3-torus. Commun. Contemp. Math. 16(4) (2014), 13500138.CrossRefGoogle Scholar
Herman, M.. Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Inst. Hautes Études Sci. Publ. Math. 49 (1979), 5–233.CrossRefGoogle Scholar
Rodriguez Hertz, F., Rodriguez Hertz, J. and Ures, R.. Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle. Invent. Math. 172(2) (2008), 353–381.CrossRefGoogle Scholar
Hirsch, M., Pugh, C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583) . Springer, Berlin, 1977.CrossRefGoogle Scholar
Rodriguez Hertz, F. and Wang, Z.. Global rigidity of higher rank abelian Anosov algebraic actions. Invent. Math. 198(1) (2014), 165–209.CrossRefGoogle Scholar
Journé, J.-L.. A regularity lemma for functions of several variables. Rev. Mat. Iberoam. 4(2) (1988), 187–193.CrossRefGoogle Scholar
Katok, A., Katok, S. and Schmidt, K.. Rigidity of measurable structure for ${\mathbb{Z}}^d$ -actions by automorphisms of a torus. Comment. Math. Helv. 77(4) (2002), 718–745.CrossRefGoogle Scholar
Katok, A. and Nitica, V.. Rigidity in Higher Rank Abelian Group Actions. Volume I. Introduction and Cocycle Problem (Cambridge Tracts in Mathematics, 185). Cambridge University Press, Cambridge, 2011.Google Scholar
Kopell, N.. Commuting diffeomorphisms. Global Analysis (Proceedings of Symposia in Pure Mathematics, XIV) . Eds. S. S. Chern and S. Smale. American Mathematical Society, Providence, RI, 1970, pp. 165–184.Google Scholar
Mañé, R.. Contributions to the stability conjecture. Topology 17(4) (1978), 383–396.CrossRefGoogle Scholar
Palis, J. and Yoccoz, J.-C.. Rigidity of centralizers of diffeomorphisms. Ann. Sci. Éc. Norm. Supér. (4) 22(1) (1989), 81–98.CrossRefGoogle Scholar
Palis, J. and Yoccoz, J.-C.. Centralizers of Anosov diffeomorphisms on tori. Ann. Sci. Éc. Norm. Supér. (4) 22(1) (1989), 99–108.CrossRefGoogle Scholar
Pinto, A. A. and Rand, D. A.. Smoothness of holonomies for codimension $1$ hyperbolic dynamics. Bull. Lond. Math. Soc. 34(3) (2002), 341–352.CrossRefGoogle Scholar
Potrie, R.. Partial hyperbolicity and foliations in ${T}^3$ . J. Mod. Dyn. 9 (2015), 81–121.CrossRefGoogle Scholar
Ponce, G., Tahzibi, A. and Varão, R., On the Bernoulli property for certain partially hyperbolic diffeomorphisms. Adv. Math. 329 (2018), 329–360.CrossRefGoogle Scholar
Rocha, J. and Varandas, P.. The centralizer of ${C}^r$ -generic diffeomorphisms at hyperbolic basic sets is trivial. Proc. Amer. Math. Soc. 146(1) (2018), 247–260.CrossRefGoogle Scholar
Sigmund, K.. Generic properties of invariant measures for Axiom A diffeomorphisms. Invent. Math. 11 (1970), 99–109.CrossRefGoogle Scholar
Smale, S.. Dynamics retrospective: great problems, attempts that failed. Nonlinear science: the next decade (Los Alamos, NM, 1990) . Phys. D 51(1–3) (1991), 267–273.CrossRefGoogle Scholar
Smale, S.. Mathematical problems for the next century. Math. Intelligencer 20(2) (1998), 7–15.CrossRefGoogle Scholar
Ures, R.. Intrinsic ergodicity of partially hyperbolic diffeomorphisms with hyperbolic linear part. Proc. Amer. Math. Soc. 140 (2012), 1973–1985.CrossRefGoogle Scholar
Viana, M. and Yang, J.. Measure-theoretical properties of center foliations. Modern Theory of Dynamical Systems (Contemporary Mathematics, 692). Eds. A. Katok, Y. Pesin and F. Rodriguez Hertz. American Mathematical Society, Providence, RI, 2017, pp. 291–320.CrossRefGoogle Scholar
Walters, P.. Topological conjugacy of affine transformations of tori. Trans. Amer. Math. Soc. 131 (1968), 40–50.CrossRefGoogle Scholar
Walters, P.. Anosov diffeomorphisms are topologically stable. Topology 9 (1970), 71–78.CrossRefGoogle Scholar