October  2021, 41(10): 4943-4957. doi: 10.3934/dcds.2021063

Symmetries of vector fields: The diffeomorphism centralizer

Department of Mathematics, The University of Chicago, Chicago, IL, 60637, USA

Received  September 2020 Revised  February 2021 Published  October 2021 Early access  March 2021

Fund Project: D.O. was supported by the ERC project 692925 NUHGD

In this paper we study the diffeomorphism centralizer of a vector field: given a vector field it is the set of diffeomorphisms that commutes with the flow. Our main theorem states that for a $ C^1 $-generic diffeomorphism having at most finitely many sinks or sources, the diffeomorphism centralizer is quasi-trivial. In certain cases, we can promote the quasi-triviality to triviality. We also obtain a criterion for a diffeomorphism in the centralizer to be a reparametrization of the flow.

Citation: Davi Obata. Symmetries of vector fields: The diffeomorphism centralizer. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4943-4957. doi: 10.3934/dcds.2021063
References:
[1]

F. AbdenurC. Bonatti and S. Crovisier, Global dominated splittings and the $C^1$-Newhouse phenomenon., Proc. Amer. Math. Soc., 134 (2006), 2229-2237.  doi: 10.1090/S0002-9939-06-08445-0.  Google Scholar

[2]

J. Alongi and G. Nelson, Recurrence and Topology, Graduate studies in mathematics, 85, American mathematical society, 2007. doi: 10.1090/gsm/085.  Google Scholar

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L. Bakker and T. Fisher, Open sets of diffeomorphisms with trivial centralizer in the $C^1$-topology, Nonlinearity, 27 (2014), 2869-2885.  doi: 10.1088/0951-7715/27/12/2869.  Google Scholar

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C. Bonatti and S. Crovisier, Récurrence et généricité., Invent. Math., 158 (2004), 33-104.  doi: 10.1007/s00222-004-0368-1.  Google Scholar

[5]

C. BonattiS. Crovisier and A. Wilkinson, The $C^1$ generic diffeomorphism has trivial centralizer, Publications Mathématiques de l'IHÉS, 109 (2009), 185-244.  doi: 10.1007/s10240-009-0021-z.  Google Scholar

[6]

C. BonattiL. Diaz and E. Pujals, A $C^1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math., 158 (2003), 355-418.  doi: 10.4007/annals.2003.158.355.  Google Scholar

[7]

C. BonattiN. Gourmelon and T. Vivier, Perturbations of the derivative along periodic orbits, Ergodic Theory Dynam. Systems, 26 (2006), 1307-1337.  doi: 10.1017/S0143385706000253.  Google Scholar

[8]

W. BonomoJ. Rocha and P. Varandas, The centralizer of Komuro-expansive flows and expansive $\Bbb R^d$ actions, Math. Z., 289 (2018), 1059-1088.  doi: 10.1007/s00209-017-1988-7.  Google Scholar

[9]

W. Bonomo and P. Varandas, $C^1$-generic sectional Axiom A flows have trivial centralizer, Port. Math., 76 (2019), 29-48.  doi: 10.4171/PM/2025.  Google Scholar

[10]

W. Bonomo and P. Varandas, A criterion for the triviality of the centralizer for vector fields and applications, J. Differential Equations, 267 (2019), 1748-1766.  doi: 10.1016/j.jde.2019.02.022.  Google Scholar

[11]

L. Burslem, Centralizers of partially hyperbolic diffeomorphisms, Ergodic Theory Dynam. Systems, 24 (2004), 55-87.  doi: 10.1017/S0143385703000191.  Google Scholar

[12]

S. Crovisier, Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms, Publications Mathématiques de l'IHÉS, 104 (2006), 87-141.  doi: 10.1007/s10240-006-0002-4.  Google Scholar

[13]

S. Crovisier and D. Yang, Homoclinic tangencies and singular hyperbolicity for three-dimensional vector fields, C. R. Math. Acad. Sci. Paris, 353 (2015), 85-88, arXiv: 1702.05994. doi: 10.1016/j.crma.2014.10.015.  Google Scholar

[14]

T. Fisher, Trivial centralizers for axiom A diffeomorphisms, Nonlinearity, 21 (2008), 2505-2517.  doi: 10.1088/0951-7715/21/11/002.  Google Scholar

[15]

T. Fisher, Trivial centralizers for codimension-one attractors, Bull. Lond. Math. Soc., 21 (2009), 51-56.  doi: 10.1112/blms/bdn100.  Google Scholar

[16]

K. Kato and A. Morimoto, Topological stability of Anosov flows and their centralizers, Topology, 12 (1973), 255-273.  doi: 10.1016/0040-9383(73)90012-8.  Google Scholar

[17]

N. Kopell, Commuting diffeomorphisms, Globa Analysis, Proc. Sympos. Pure Math., XIV (1970), 165-184.   Google Scholar

[18]

M. LeguilD. Obata and B. Santiago, On the centralizer of vector fields: Criteria of triviality and genericity results, Math. Z., 297 (2021), 283-337.  doi: 10.1007/s00209-020-02511-x.  Google Scholar

[19]

I. Mundet I Riera, Automorphisms of generic gradient vector fields with prescribed finite symmetries, Revista Matemática Iberoamericana, 35 (2019), 1281-1308.  doi: 10.4171/rmi/1083.  Google Scholar

[20]

M. Oka, Expansive flows and their centralizers,, Nagoya Math. J., 64 (1976), 1-15.  doi: 10.1017/S0027763000017517.  Google Scholar

[21]

J. Palis and W. de Melo, Geometric Theory of Dynamical Systems, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[22]

J. Palis and J. C. Yoccoz, Centralizers of {A}nosov diffeomorphisms on tori, Ann. Sci. École Norm. Sup. (4), 22 (1989), 98-108. doi: 10.24033/asens. 1577.  Google Scholar

[23]

J. Palis and J. C. Yoccoz, Rigidity of centralizers of diffeomorphisms, Ann. Sci. École Norm. Sup. (4), 22 (1989), 81-98. doi: 10.24033/asens. 1576.  Google Scholar

[24]

M. Peixoto, Structural stability on two-dimensional manifolds, Bol. Soc. Mat. Mexicana (2), 5 (1960), 188-189.   Google Scholar

[25]

P. Sad, Centralizers of vector fields, Topology, 18 (1979), 97-104.  doi: 10.1016/0040-9383(79)90027-2.  Google Scholar

[26]

S. Smale, Dynamics retrospective: Great problems, attempts that failed, Nonlinear Science: The Next Decade, Los Alamos, NM, 1990, Physica D., 51 (1991), 267-273.  doi: 10.1016/0167-2789(91)90238-5.  Google Scholar

[27]

S. Smale, Mathematical problems for the next century, Math. Intelligencer, 20 (1998), 7-15.  doi: 10.1007/BF03025291.  Google Scholar

[28]

R. Thom, Sur Les Intégrales Premières d'un Système Différentiel sur une Variété Compacte, Unpublished manuscript. Google Scholar

show all references

References:
[1]

F. AbdenurC. Bonatti and S. Crovisier, Global dominated splittings and the $C^1$-Newhouse phenomenon., Proc. Amer. Math. Soc., 134 (2006), 2229-2237.  doi: 10.1090/S0002-9939-06-08445-0.  Google Scholar

[2]

J. Alongi and G. Nelson, Recurrence and Topology, Graduate studies in mathematics, 85, American mathematical society, 2007. doi: 10.1090/gsm/085.  Google Scholar

[3]

L. Bakker and T. Fisher, Open sets of diffeomorphisms with trivial centralizer in the $C^1$-topology, Nonlinearity, 27 (2014), 2869-2885.  doi: 10.1088/0951-7715/27/12/2869.  Google Scholar

[4]

C. Bonatti and S. Crovisier, Récurrence et généricité., Invent. Math., 158 (2004), 33-104.  doi: 10.1007/s00222-004-0368-1.  Google Scholar

[5]

C. BonattiS. Crovisier and A. Wilkinson, The $C^1$ generic diffeomorphism has trivial centralizer, Publications Mathématiques de l'IHÉS, 109 (2009), 185-244.  doi: 10.1007/s10240-009-0021-z.  Google Scholar

[6]

C. BonattiL. Diaz and E. Pujals, A $C^1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math., 158 (2003), 355-418.  doi: 10.4007/annals.2003.158.355.  Google Scholar

[7]

C. BonattiN. Gourmelon and T. Vivier, Perturbations of the derivative along periodic orbits, Ergodic Theory Dynam. Systems, 26 (2006), 1307-1337.  doi: 10.1017/S0143385706000253.  Google Scholar

[8]

W. BonomoJ. Rocha and P. Varandas, The centralizer of Komuro-expansive flows and expansive $\Bbb R^d$ actions, Math. Z., 289 (2018), 1059-1088.  doi: 10.1007/s00209-017-1988-7.  Google Scholar

[9]

W. Bonomo and P. Varandas, $C^1$-generic sectional Axiom A flows have trivial centralizer, Port. Math., 76 (2019), 29-48.  doi: 10.4171/PM/2025.  Google Scholar

[10]

W. Bonomo and P. Varandas, A criterion for the triviality of the centralizer for vector fields and applications, J. Differential Equations, 267 (2019), 1748-1766.  doi: 10.1016/j.jde.2019.02.022.  Google Scholar

[11]

L. Burslem, Centralizers of partially hyperbolic diffeomorphisms, Ergodic Theory Dynam. Systems, 24 (2004), 55-87.  doi: 10.1017/S0143385703000191.  Google Scholar

[12]

S. Crovisier, Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms, Publications Mathématiques de l'IHÉS, 104 (2006), 87-141.  doi: 10.1007/s10240-006-0002-4.  Google Scholar

[13]

S. Crovisier and D. Yang, Homoclinic tangencies and singular hyperbolicity for three-dimensional vector fields, C. R. Math. Acad. Sci. Paris, 353 (2015), 85-88, arXiv: 1702.05994. doi: 10.1016/j.crma.2014.10.015.  Google Scholar

[14]

T. Fisher, Trivial centralizers for axiom A diffeomorphisms, Nonlinearity, 21 (2008), 2505-2517.  doi: 10.1088/0951-7715/21/11/002.  Google Scholar

[15]

T. Fisher, Trivial centralizers for codimension-one attractors, Bull. Lond. Math. Soc., 21 (2009), 51-56.  doi: 10.1112/blms/bdn100.  Google Scholar

[16]

K. Kato and A. Morimoto, Topological stability of Anosov flows and their centralizers, Topology, 12 (1973), 255-273.  doi: 10.1016/0040-9383(73)90012-8.  Google Scholar

[17]

N. Kopell, Commuting diffeomorphisms, Globa Analysis, Proc. Sympos. Pure Math., XIV (1970), 165-184.   Google Scholar

[18]

M. LeguilD. Obata and B. Santiago, On the centralizer of vector fields: Criteria of triviality and genericity results, Math. Z., 297 (2021), 283-337.  doi: 10.1007/s00209-020-02511-x.  Google Scholar

[19]

I. Mundet I Riera, Automorphisms of generic gradient vector fields with prescribed finite symmetries, Revista Matemática Iberoamericana, 35 (2019), 1281-1308.  doi: 10.4171/rmi/1083.  Google Scholar

[20]

M. Oka, Expansive flows and their centralizers,, Nagoya Math. J., 64 (1976), 1-15.  doi: 10.1017/S0027763000017517.  Google Scholar

[21]

J. Palis and W. de Melo, Geometric Theory of Dynamical Systems, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[22]

J. Palis and J. C. Yoccoz, Centralizers of {A}nosov diffeomorphisms on tori, Ann. Sci. École Norm. Sup. (4), 22 (1989), 98-108. doi: 10.24033/asens. 1577.  Google Scholar

[23]

J. Palis and J. C. Yoccoz, Rigidity of centralizers of diffeomorphisms, Ann. Sci. École Norm. Sup. (4), 22 (1989), 81-98. doi: 10.24033/asens. 1576.  Google Scholar

[24]

M. Peixoto, Structural stability on two-dimensional manifolds, Bol. Soc. Mat. Mexicana (2), 5 (1960), 188-189.   Google Scholar

[25]

P. Sad, Centralizers of vector fields, Topology, 18 (1979), 97-104.  doi: 10.1016/0040-9383(79)90027-2.  Google Scholar

[26]

S. Smale, Dynamics retrospective: Great problems, attempts that failed, Nonlinear Science: The Next Decade, Los Alamos, NM, 1990, Physica D., 51 (1991), 267-273.  doi: 10.1016/0167-2789(91)90238-5.  Google Scholar

[27]

S. Smale, Mathematical problems for the next century, Math. Intelligencer, 20 (1998), 7-15.  doi: 10.1007/BF03025291.  Google Scholar

[28]

R. Thom, Sur Les Intégrales Premières d'un Système Différentiel sur une Variété Compacte, Unpublished manuscript. Google Scholar

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