Let F be a C ∞ vector field defined near the origin O ∈ ℝn, F(O) = 0, and (Ft) be its local flow. Denote by the set of germs of orbit preserving diffeomorphisms h: ℝn → ℝn at O, and let , (r ≥ 0), be the identity component of with respect to the weak Whitney Wr topology. Then contains a subset consisting of maps of the form Fα(x)(x), where α: ℝn → ℝ runs over the space of all smooth germs at O. It was proved earlier by the author that if F is a linear vector field, then = . In this paper we present a class of examples of vector fields with degenerate singularities at O for which formally coincides with , i.e. on the level of ∞-jets at O. We also establish parameter rigidity of linear vector fields and “reduced” Hamiltonian vector fields of real homogeneous polynomials in two variables.
@article{bwmeta1.element.doi-10_2478_s11533-009-0010-y,
author = {Sergiy Maksymenko},
title = {$\infty$-jets of diffeomorphisms preserving orbits of vector fields},
journal = {Open Mathematics},
volume = {7},
year = {2009},
pages = {272-298},
zbl = {1187.37028},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-009-0010-y}
}
Sergiy Maksymenko. ∞-jets of diffeomorphisms preserving orbits of vector fields. Open Mathematics, Tome 7 (2009) pp. 272-298. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-009-0010-y/
[1] Abe K., Fukui K., On the structure of the groups of equivariant diffeomorphisms of G-manifolds with codimension one orbit, Topology, 2001, 40, 1325–1337 http://dx.doi.org/10.1016/S0040-9383(00)00014-8[Crossref] | Zbl 1001.58003
[2] Abe K., Fukui K., On the first homology of automorphism groups of manifolds with geometric structures, Cent. Eur. J. Math., 2005, 3, 516–528 http://dx.doi.org/10.2478/BF02475921[Crossref] | Zbl 1117.57032
[3] Banyaga A., On the structure of the group of equivariant diffeomorphisms, Topology, 1977, 16, 279–283 http://dx.doi.org/10.1016/0040-9383(77)90009-X[Crossref]
[4] Camacho C., Neto A.L., Orbit preserving diffeomorphisms and the stability of Lie group actions and singular foliations, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976), 82–103, Lecture Notes in Math., 597, Springer, Berlin, 1977
[5] Chacon R.V., Change of velocity in flows, J. Math. Mech., 1966, 16, 417–431 | Zbl 0158.30301
[6] Damjanović D., Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions, J. Mod. Dyn., 2007, 1, 665–688 [WoS][Crossref] | Zbl 1139.37017
[7] Einsiedler M., Fisher T., Differentiable rigidity for hyperbolic toral actions, Israel J. Math., 2007, 157, 347–377 http://dx.doi.org/10.1007/s11856-006-0016-0[Crossref] | Zbl 1112.37022
[8] Golubitsky M., Guillemin V., Stable mappings and their singularities, Graduate Texts in Mathematics, 14, Springer-Verlag, 1973 | Zbl 0294.58004
[9] Gutiérrez C., De Melo W., The connected components of Morse-Smale vector fields on two manifolds, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976), 230–251, Lecture Notes in Math., 597, Springer, Berlin, 1977
[10] Hart D., On the smoothness of generators, Topology, 1983, 22, 357–363 http://dx.doi.org/10.1016/0040-9383(83)90021-6[Crossref]
[11] Hirsch M., Differential topology, Graduate Texts in Mathematics, 33, Springer-Verlag, 1976 | Zbl 0356.57001
[12] Hopf E., Ergodentheorie, Berlin, 1937 (in German)
[13] Hurder S., A survey of rigidity theory for Anosov actions, In: Differential topology, foliations, and group actions, Rio de Janeiro, 1992, Contemp. Math., 161, Amer. Math. Soc., Providence, 1994
[14] Kanai M., A new approach to the rigidity of discrete group actions, Geom. Funct. Anal., 1996, 6, 943–1056 http://dx.doi.org/10.1007/BF02246995[Crossref] | Zbl 0874.58006
[15] Katok A., Spatzier R.J., First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity, Inst. Hautes Études Sci. Publ. Math., 1994, 79, 131–156 http://dx.doi.org/10.1007/BF02698888[Crossref] | Zbl 0819.58027
[16] Katok A., Spatzier R.J., Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions, Proc. Steklov Inst. Math., 1997, 216, 287–314 | Zbl 0938.37010
[17] Kochergin A.V., Change of time in flows and mixing, Izv. Akad. Nauk SSSR Ser. Mat., 1973, 37, 1275–1298 (in Russian) | Zbl 0286.28013
[18] Kowada M., The orbit-preserving transformation groups associated with a measurable flow, J. Math. Soc. Japan, 1972, 24, 355–373 http://dx.doi.org/10.2969/jmsj/02430355[Crossref]
[19] Maksymenko S., Smooth shifts along trajectories of flows, Topology Appl., 2003, 130, 183–204 http://dx.doi.org/10.1016/S0166-8641(02)00363-2[WoS][Crossref] | Zbl 1014.37012
[20] Maksymenko S., Homotopy types of stabilizers and orbits of Morse functions on surfaces, Ann. Global Anal. Geom., 2006, 29, 241–285 http://dx.doi.org/10.1007/s10455-005-9012-6[Crossref] | Zbl 1099.37013
[21] Maksymenko S., Stabilizers and orbits of smooth functions, Bull. Sci. Math., 2006, 130, 279–311 http://dx.doi.org/10.1016/j.bulsci.2005.11.001[Crossref] | Zbl 1109.37023
[22] Maksymenko S., Hamiltonian vector fields of homogeneous polynomials on the plane, Topological problems and relative questions, Proceedings of Institute of Mathematics of Ukrainian NAS, 2006, 3, 269–308 (in Ukrainian)
[23] Mather J., Differentiable invariants, Topology, 1977, 16, 145–155 http://dx.doi.org/10.1016/0040-9383(77)90012-X[Crossref]
[24] Matsumoto Sh., Mitsumatsu Y., Leafwise cohomology and rigidity of certain Lie group actions, Ergodic Theory Dynam. Systems, 2003, 23, 1839–1866 http://dx.doi.org/10.1017/S0143385703000038[Crossref] | Zbl 1052.22020
[25] Ornstein D.S., Smorodinsky M., Continuous speed changes for flows, Israel J. Math., 1978, 31, 161–168 http://dx.doi.org/10.1007/BF02760547[Crossref] | Zbl 0399.58003
[26] Palis J., de Melo W., Geometric theory of dynamical systems, Springer-Verlag, New York-Berlin, 1982 | Zbl 0491.58001
[27] Parry W., Cocycles and velocity changes, J. London Math. Soc., 1972, 5, 511–516 http://dx.doi.org/10.1112/jlms/s2-5.3.511[Crossref]
[28] Rybicki T., Isomorphisms between leaf preserving diffeomorphism groups, Soochow J. Math., 1996, 22, 525–542 | Zbl 0869.58004
[29] Rybicki T., Homology of the group of leaf preserving homeomorphisms, Demonstratio Math., 1996, 29, 459–464 | Zbl 0901.57030
[30] Rybicki T., On commutators of equivariant homeomorphisms, Topology Appl., 2007, 154, 1561–1564 http://dx.doi.org/10.1016/j.topol.2006.12.003[Crossref][WoS]
[31] Schwarz G.W., Smooth functions invariant under the action of a compact Lie group, Topology, 1975, 14, 63–68 http://dx.doi.org/10.1016/0040-9383(75)90036-1[Crossref]
[32] Schwarz G.W., Lifting smooth homotopies of orbit spaces, Inst. Hautes Études Sci. Publ. Math., 1980, 51, 37–135 http://dx.doi.org/10.1007/BF02684776[Crossref] | Zbl 0449.57009
[33] dos Santos N.M., Parameter rigid actions of the Heisenberg groups, Ergodic Theory Dynam. Systems, 2007, 27, 1719–1735 http://dx.doi.org/10.1017/S014338570600109X[Crossref] | Zbl 1127.37026
[34] Siegel C.L., Über die Normalform analytischer Differentialgleichungen in der Nähe einer Gleichgewichtslösung, Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. Math.-Phys.-Chem. Abt., 1952, 21–30 | Zbl 0047.32901
[35] Sternberg S., Local contractions and a theorem of Poincaré, Amer. J. Math., 1957, 79, 809–824 http://dx.doi.org/10.2307/2372437[Crossref]
[36] Totoki H., Time changes of flows, Mem. Fac. Sci. Kyushu Univ. Ser. A, 1966, 20, 27–55 | Zbl 0185.39103
[37] Venti R., Linear normal forms of differential equations, J. Differential Equations, 1966, 2, 182–194 http://dx.doi.org/10.1016/0022-0396(66)90042-8[Crossref] | Zbl 0135.30902