Centralizers of hyperbolic and kinematic-expansive flows
Mathematics Research Reports, Volume 2 (2021), pp. 21-44.

We show that generic C hyperbolic flows commute with no C -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.

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DOI: 10.5802/mrr.8
Classification: 37D20, 37C10, 37C20
Keywords: Dynamical systems; flows; commuting; expansive; hyperbolic
Bakker, Lennard 1; Fisher, Todd 1; Hasselblatt, Boris 2

1 Department of Mathematics, Brigham Young University, Provo, UT 84602, USA
2 Department of Mathematics, Tufts University, Medford, MA 02144, USA
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Bakker, Lennard; Fisher, Todd; Hasselblatt, Boris. Centralizers of hyperbolic and kinematic-expansive flows. Mathematics Research Reports, Volume 2 (2021), pp. 21-44. doi : 10.5802/mrr.8. http://www.numdam.org/articles/10.5802/mrr.8/

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

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

[3] Bakker, Lennard; Fisher, Todd 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] Bonatti, Christian; Crovisier, Sylvain; Vago, Gioia M.; Wilkinson, Amie 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] Bonatti, Christian; Crovisier, Sylvain; Wilkinson, Amie C 1 -generic conservative diffeomorphisms have trivial centralizer, J. Mod. Dyn., Volume 2 (2008) no. 2, pp. 359-373 | DOI | MR | Zbl

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

[7] Bonatti, Christian; Crovisier, Sylvain; Wilkinson, Amie 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] Bonomo, Wescley; Rocha, Jorge; Varandas, Paulo The centralizer of Komuro-expansive flows and expansive d actions, Math. Z., Volume 289 (2018) no. 3-4, pp. 1059-1088 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

[23] Obata, Davi Joel dos Anjos 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] Oka, Masatoshi Expansive flows and their centralizers, Nagoya Math. J., Volume 64 (1976), pp. 1-15 | MR | Zbl

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

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

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

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

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

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

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