Centralizers of hyperbolic and kinematic-expansive flows

Lennard Bakker; Todd Fisher; Boris Hasselblatt

doi:10.5802/mrr.8

Lennard Bakker ¹; Todd Fisher ¹; Boris Hasselblatt ²

¹ Department of Mathematics, Brigham Young University, Provo, UT 84602, USA
² Department of Mathematics, Tufts University, Medford, MA 02144, USA

Mathematics Research Reports, Volume 2 (2021), pp. 21-44.

Abstract

We show that generic $C^{\infty}$ hyperbolic flows commute with no $C^{\infty}$ -diffeomorphism other than a time- $t$ map of the flow itself. Kinematic-expansivity, a substantial weakening of expansivity, implies that $C^{0}$ flows have quasidiscrete $C^{0}$ -centralizer, and additional conditions broader than transitivity then give discrete $C^{0}$ -centralizer. We also prove centralizer-rigidity: a diffeomorphism commuting with a generic hyperbolic flow is determined by its values on any open set.

Received: 2019-03-26
Revised: 2021-05-11
Published online: 2021-09-03

DOI: 10.5802/mrr.8

Classification: 37D20, 37C10, 37C20
Keywords: Dynamical systems; flows; commuting; expansive; hyperbolic

Author's affiliations:

Lennard Bakker ¹; Todd Fisher ¹; Boris Hasselblatt ²

¹ Department of Mathematics, Brigham Young University, Provo, UT 84602, USA
² Department of Mathematics, Tufts University, Medford, MA 02144, USA

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Lennard Bakker; Todd Fisher; Boris Hasselblatt. Centralizers of hyperbolic and kinematic-expansive flows. Mathematics Research Reports, Volume 2 (2021), pp. 21-44. doi : 10.5802/mrr.8. https://mrr.centre-mersenne.org/articles/10.5802/mrr.8/

References
Cited by

[1] Boyd Anderson Diffeomorphisms with discrete centralizer, Topology, Volume 15 (1976) no. 2, pp. 143-147 | DOI | MR | Zbl

[2] Alfonso Artigue Kinematic expansive flows, Ergodic Theory Dynam. Systems, Volume 36 (2016) no. 2, pp. 390-421 | DOI | MR | Zbl

[3] Lennard Bakker; Todd Fisher Open sets of diffeomorphisms with trivial centralizer in the $C^{1}$ topology, Nonlinearity, Volume 27 (2014) no. 12, pp. 2869-2885 | DOI | MR | Zbl

[4] Christian Bonatti; Sylvain Crovisier; Gioia M. Vago; Amie Wilkinson Local density of diffeomorphisms with large centralizers, Ann. Sci. Éc. Norm. Supér. (4), Volume 41 (2008) no. 6, pp. 925-954 | DOI | Numdam | MR | Zbl

[5] Christian Bonatti; Sylvain Crovisier; Amie Wilkinson $C^{1}$ -generic conservative diffeomorphisms have trivial centralizer, J. Mod. Dyn., Volume 2 (2008) no. 2, pp. 359-373 | DOI | MR | Zbl

[6] Christian Bonatti; Sylvain Crovisier; Amie Wilkinson The centralizer of a $C^{1}$ -generic diffeomorphism is trivial, Electron. Res. Announc. Math. Sci., Volume 15 (2008), pp. 33-43 | MR | Zbl

[7] Christian Bonatti; Sylvain Crovisier; Amie Wilkinson The $C^{1}$ generic diffeomorphism has trivial centralizer, Publ. Math. Inst. Hautes Études Sci. (2009) no. 109, pp. 185-244 | DOI | Numdam | MR | Zbl

[8] Wescley Bonomo; Jorge Rocha; Paulo Varandas The centralizer of Komuro-expansive flows and expansive $ℝ^{d}$ actions, Math. Z., Volume 289 (2018) no. 3-4, pp. 1059-1088 | DOI | MR | Zbl

[9] Wescley Bonomo; Paulo Varandas A criterion for the triviality of the centralizer for vector fields and applications, J. Differential Equations, Volume 267 (2019) no. 3, pp. 1748-1766 | DOI | MR | Zbl

[10] Rufus Bowen; Peter Walters Expansive one-parameter flows, Journal of Differential Equations, Volume 12 (1972), pp. 180-193 | DOI | MR | Zbl

[11] Todd Fisher Trivial centralizers for Axiom A diffeomorphisms, Nonlinearity, Volume 21 (2008) no. 11, pp. 2505-2517 | DOI | MR | Zbl

[12] Todd Fisher; Boris Hasselblatt Hyperbolic flows, Zürich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2019 (http://www.ms.u-tokyo.ac.jp/lecturenotes16-hasselblatt.pdf) | Zbl

[13] Todd Fisher; Boris Hasselblatt Accessibility and centralizers for partially hyperbolic flows, Ergodic Theory and Dynamical Systems, Volume 42 (2022) (to appear)

[14] Etienne Ghys Flots d’Anosov dont les feuilletages stables sont différentiables, Annales Scientifiques de l’Ecole Normale Supérieure. Quatrième Série, Volume 20 (1987) no. 2, pp. 251-270 | DOI | Zbl

[15] A. A. Gura Separating diffeomorphisms of a torus, Mat. Zametki, Volume 18 (1975) no. 1, pp. 41-49 | MR | Zbl

[16] A. A. Gura The horocycle flow on a surface of negative curvature is separating, Mat. Zametki, Volume 36 (1984) no. 2, pp. 279-284 | MR

[17] Anatole Katok; Viorel Niţică Rigidity in higher rank abelian group actions. Volume I, Cambridge Tracts in Mathematics, 185, Cambridge University Press, Cambridge, 2011 | DOI | MR | Zbl

[18] Anatole Katok; Ralf J. Spatzier First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity, Inst. Hautes Études Sci. Publ. Math. (1994) no. 79, pp. 131-156 | DOI | Numdam | MR | Zbl

[19] Anatole Katok; Ralf J. Spatzier Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions, Trudy Matematicheskogo Instituta Imeni V. A. Steklova. Rossiĭskaya Akademiya Nauk, Volume 216 (1997) no. Din. Sist. i Smezhnye Vopr., pp. 292-319

[20] Nancy Kopell Commuting diffeomorphisms, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 165-184 | MR | Zbl

[21] Martin Leguil; Davi Obata; Bruno Santiago On the centralizer of vector fields: criteria of triviality and genericity results, Mathematische Zeitschrift, Volume 297 (2021), pp. 283-337 | DOI | MR | Zbl

[22] Shigenori Matsumoto Kinematic expansive suspensions of irrational rotations on the circle, Hokkaido Math. J., Volume 46 (2017) no. 3, pp. 473-485 | DOI | MR | Zbl

[23] Davi Joel dos Anjos Obata Symmetries of vector fields: the diffeomorphism centralizer (arXiv:1903.05883, see also https://www.imo.universite-paris-saclay.fr/~obata/Tese-Ufrj-Davi.pdf)

[24] Masatoshi Oka Expansive flows and their centralizers, Nagoya Math. J., Volume 64 (1976), pp. 1-15 | MR | Zbl

[25] J. Palis Rigidity of the centralizers of diffeomorphisms and structural stability of suspended foliations, Differential topology, foliations and Gelfand-Fuks cohomology (Proc. Sympos., Pontifícia Univ. Católica, Rio de Janeiro, 1976) (Lecture Notes in Math.), Volume 652, Springer, Berlin, 1978, pp. 114-121 | DOI | MR | Zbl

[26] Jacob Palis; Jean-Christophe Yoccoz Rigidity of centralizers of diffeomorphisms, Ann. Sci. École Norm. Sup. (4), Volume 22 (1989) no. 1, pp. 81-98 | DOI | Numdam | MR | Zbl

[27] Jorge Rocha A note on the $C^{0}$ -centralizer of an open class of bidimensional Anosov diffeomorphisms, Aequationes Math., Volume 76 (2008) no. 1-2, pp. 105-111 | DOI | MR | Zbl

[28] Jorge Rocha; Paulo Varandas The centralizer of $C^{r}$ -generic diffeomorphisms at hyperbolic basic sets is trivial, Proc. Amer. Math. Soc., Volume 146 (2018) no. 1, pp. 247-260 | DOI | MR | Zbl

[29] Federico Rodriguez Hertz; Zhiren Wang Global rigidity of higher rank abelian Anosov algebraic actions, Invent. Math., Volume 198 (2014) no. 1, pp. 165-209 | DOI | MR | Zbl

[30] Paulo Roberto Sad Centralizers of vector fields, Topology, Volume 18 (1979) no. 2, pp. 97-104 | DOI | MR | Zbl

[31] George R. Sell Smooth linearization near a fixed point, Amer. J. Math., Volume 107 (1985) no. 5, pp. 1035-1091 | DOI | MR | Zbl

[32] Steve Smale Mathematical problems for the next century, Math. Intelligencer, Volume 20 (1998) no. 2, pp. 7-15 | DOI | MR | Zbl

[33] Peter Walters Homeomorphisms with discrete centralizers and ergodic properties, Math. Systems Theory, Volume 4 (1970), pp. 322-326 | DOI | MR | Zbl

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