𝑆𝑂(2)-equivariant vector fields on 3-manifolds: moduli of stability and genericity

dos Reis, Genesio Lima; dos Santos, Geovan Tavares

doi:10.1090/S0002-9947-1988-0951626-7

$\textrm {SO}(2)$-equivariant vector fields on $3$-manifolds: moduli of stability and genericity
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by Genesio Lima dos Reis and Geovan Tavares dos Santos

Trans. Amer. Math. Soc. 308 (1988), 747-763

DOI: https://doi.org/10.1090/S0002-9947-1988-0951626-7

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Abstract:

An open and dense class of vector fields on $3$-dimensional compact manifolds equivariant under the action of $\operatorname {SO} (2)$ is defined. Each such vector field has finite moduli of stability. We also exhibit an open and dense subset of the $\operatorname {SO} (2)$-equivariant gradient vector fields which are structurally stable.

References

Edward Bierstone, General position of equivariant maps, Trans. Amer. Math. Soc. 234 (1977), no. 2, 447–466. MR 464287, DOI 10.1090/S0002-9947-1977-0464287-3
Raoul Bott, Nondegenerate critical manifolds, Ann. of Math. (2) 60 (1954), 248–261. MR 64399, DOI 10.2307/1969631
Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
Mike Field, Equivariant dynamical systems, Bull. Amer. Math. Soc. 76 (1970), 1314–1318. MR 277850, DOI 10.1090/S0002-9904-1970-12657-X
M. J. Field, Equivariant dynamical systems, Trans. Amer. Math. Soc. 259 (1980), no. 1, 185–205. MR 561832, DOI 10.1090/S0002-9947-1980-0561832-4
Carlos Gutiérrez, Structural stability for flows on the torus with a cross-cap, Trans. Amer. Math. Soc. 241 (1978), 311–320. MR 492303, DOI 10.1090/S0002-9947-1978-0492303-2
Philip Hartman, On local homeomorphisms of Euclidean spaces, Bol. Soc. Mat. Mexicana (2) 5 (1960), 220–241. MR 141856
M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173, DOI 10.1007/BFb0092042
Nelson G. Markley, The Poincaré-Bendixson theorem for the Klein bottle, Trans. Amer. Math. Soc. 135 (1969), 159–165. MR 234442, DOI 10.1090/S0002-9947-1969-0234442-1

Bifurcações dos campos vetoriais equivariantes

W. de Melo, G. L. dos Reis, and P. Mendes, Equivariant diffeomorphisms with simple recurrences on two-manifolds, Trans. Amer. Math. Soc. 289 (1985), no. 2, 793–807. MR 784014, DOI 10.1090/S0002-9947-1985-0784014-4
Peter Orlik and Frank Raymond, Actions of $\textrm {SO}(2)$ on 3-manifolds, Proc. Conf. on Transformation Groups (New Orleans, La., 1967) Springer, New York, 1968, pp. 297–318. MR 0263112
J. Palis, On Morse-Smale dynamical systems, Topology 8 (1968), 385–404. MR 246316, DOI 10.1016/0040-9383(69)90024-X

A differentiable invariant of topological conjugacies and moduli of stability

49/50

M. J. Pacífico, Structural stability of vector fields on $3$-manifolds with boundary, J. Differential Equations 54 (1984), no. 3, 346–372. MR 760376, DOI 10.1016/0022-0396(84)90148-7
M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology 1 (1962), 101–120. MR 142859, DOI 10.1016/0040-9383(65)90018-2

On the classification of flows on

manifolds

Jacob Palis Jr. and Welington de Melo, Geometric theory of dynamical systems, Springer-Verlag, New York-Berlin, 1982. An introduction; Translated from the Portuguese by A. K. Manning. MR 669541, DOI 10.1007/978-1-4612-5703-5
Charles C. Pugh, The closing lemma, Amer. J. Math. 89 (1967), 956–1009. MR 226669, DOI 10.2307/2373413
G. L. dos Reis, Structural stability of equivariant vector fields on two-manifolds, Trans. Amer. Math. Soc. 283 (1984), no. 2, 633–643. MR 737889, DOI 10.1090/S0002-9947-1984-0737889-8
Gerald W. Schwarz, Lifting smooth homotopies of orbit spaces, Inst. Hautes Études Sci. Publ. Math. 51 (1980), 37–135. MR 573821, DOI 10.1007/BF02684776
S. Smale, The $\Omega$-stability theorem, Global Analysis (Proc. Sympos. Pure Math., Vols. XIV, XV, XVI, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 289–297. MR 0271971
Shlomo Sternberg, On the structure of local homeomorphisms of euclidean $n$-space. II, Amer. J. Math. 80 (1958), 623–631. MR 96854, DOI 10.2307/2372774

One parameter families of vector fields—Bifurcations near saddle-connections

Floris Takens, Partially hyperbolic fixed points, Topology 10 (1971), 133–147. MR 307279, DOI 10.1016/0040-9383(71)90035-8
Arthur G. Wasserman, Equivariant differential topology, Topology 8 (1969), 127–150. MR 250324, DOI 10.1016/0040-9383(69)90005-6