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Poincaré-Hopf index and partial hyperbolicity
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 17 (2008) no. 1, pp. 193-206.

We use the theory of partially hyperbolic systems [HPS] in order to find singularities of index 1 for vector fields with isolated zeroes in a 3-ball. Indeed, we prove that such zeroes exists provided the maximal invariant set in the ball is partially hyperbolic, with volume expanding central subbundle, and the strong stable manifolds of the singularities are unknotted in the ball.

Nous utilisons des systèmes partiellement hyperboliques [HPS] pour trouver des singularités d’indice 1 pour les champs de vecteurs avec singularités isolées sur la boule tridimensionelle. En fait, on trouvera de telles singularités lorsque l’ensemble maximal invariant dans la boule est partiellement hyperbolique, à sous-fibré central volume-dilatant, et les variétés stables fortes sur les singularités sont toutes non nouées.

DOI: 10.5802/afst.1180
C. A Morales 1

1 Instituto de Matematica, Universidade Federal do Rio de Janeiro, P. O. Box 68530, 21945-970 Rio de Janeiro, Brazil.
     author = {C. A Morales},
     title = {Poincar\'e-Hopf index and partial hyperbolicity},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {193--206},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 17},
     number = {1},
     year = {2008},
     doi = {10.5802/afst.1180},
     mrnumber = {2464098},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1180/}
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PB  - Université Paul Sabatier, Institut de Mathématiques
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C. A Morales. Poincaré-Hopf index and partial hyperbolicity. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 17 (2008) no. 1, pp. 193-206. doi : 10.5802/afst.1180. https://afst.centre-mersenne.org/articles/10.5802/afst.1180/

[ABS] Afraimovich (V.S.), Bykov (V. V.), Shilnikov (L. P.).— On attracting structurally unstable limit sets of Lorenz attractor type (Russian) Trudy Moskov. Mat. Obshch. 44, 150-212 (1982). | MR | Zbl

[B] Bautista (S.).— Sobre conjuntos singulares-hiperbólicos, Thesis Universidade Federal do Rio de Janeiro (2005).

[BMo] Bautista (S.), Morales (C.).— Existence of periodic orbits for singular-hyperbolic sets, Mosc. Math. J. 6, no. 2, 265-297 (2006). | MR | Zbl

[BM] Bing (R. H.), Martin (J. M.), Cubes with knotted holes, Trans. Amer. Math. Soc. 155, 217-231 (1971). | MR | Zbl

[BDV] Bonatti (C.), Diaz (L.), Viana (M.).— Dynamics beyond uniform hyperbolicity. A global geometric and probabilistic perspective, Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics, III. Springer-Verlag, Berlin, 2005. | MR | Zbl

[Br] Brunella (M.).— Separating basic sets of a nontransitive Anosov flow, Bull. London Math. Soc. 25, 487-490 (1993). | MR | Zbl

[CMV] Cima (A.), Mañosas (F.), Villadelprat (J.).— A Poincaré-Hopf theorem for noncompact manifolds, (English. English summary) Topology 37, no. 2, 261-277 (1998). | MR | Zbl

[C] Conley (C.).— Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, 38. American Mathematical Society, Providence, R.I., 1978. | MR | Zbl

[DO] Dancer (E. N.), Ortega (R.).— The index of Lyapunov stable fixed points in two dimensions, (English. English summary) J. Dynam. Differential Equations 6, no. 4, 631-637 (1994). | MR | Zbl

[E] Eliashberg (Ya. M.).— Combinatorial methods in symplectic geometry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 531-539, Amer. Math. Soc., Providence, RI, 1987. | MR | Zbl

[F] Franks (J.).— Rotation vectors and fixed points of area preserving surface diffeomorphisms, (English. English summary) Trans. Amer. Math. Soc. 348, no. 7, 2637-2662 (1996). | MR | Zbl

[G] Gabai (D.).— 3 lectures on foliations and laminations on 3-manifolds, Laminations and foliations in dynamics, geometry and topology (Stony Brook, NY, 1998), 87-109, Contemp. Math., 269, Amer. Math. Soc., Providence, RI, 2001. | MR | Zbl

[GT] Gilmore (R.), Tsankov (T.).— Topological aspects of the structure of chaotic attractors in 3 , (English. English summary) Phys. Rev. E (3) 69 (2004), no. 5, 056206, 11 pp. | MR

[GW] Guckenheimer (J.), Williams (R.).— Structural stability of Lorenz attractors, Publ Math IHES 50, 59-72 (1979). | Numdam | MR | Zbl

[GMZ] Grines (V. Z.), Medvedev (V. S.), Zhuzhoma (E. V.).— New relations for Morse-Smale systems with trivially embedded one-dimensional separatrices, (Russian) Mat. Sb. 194, no. 7, 25-56; translation in Sb. Math. 194 (2003), no. 7-8, 979-1007 (2003). | MR | Zbl

[HK] Hasselblatt (B.), Katov (A.).— Introduction to the modern theory of dynamical systems, Cambridge University Press, Cambridge (1995). | MR | Zbl

[HPS] Hirsch (M.), Pugh (C.), Shub (M.).— Invariant manifolds, Lec. Not. in Math. 583 (1977), Springer-Verlag. | MR | Zbl

[L] Le Calvez (P.).— Une propriété dynamique des homéomorphismes du plan au voisinage d’un point fixe d’indice >1 (French. English, French summary) [A dynamical property of homeomorphisms of the plane in the neighborhood of a fixed point of index >1] Topology 38, no. 1, 23-35 (1999). | Zbl

[M] Matsumoto (S.).— Arnold conjecture for surface homeomorphisms (English. English summary) Proceedings of the French-Japanese Conference “Hyperspace Topologies and Applications" (La Bussiére, 1997). Topology Appl. 104, no. 1-3, 191-214 (2000). | Zbl

[Mi] Milnor (J.).— Topology from the differentiable viewpoint, Based on notes by David W. Weaver The University Press of Virginia, Charlottesville, Va. 1965. | MR | Zbl

[M1] Morales (C.).— Examples of singular-hyperbolic attracting sets, Dyn. Syst. (To appear). | MR | Zbl

[MPP1] Morales (C.), Pacifico (M. J.), Pujals (E. R.).— Strange attractors across the boundary of hyperbolic systems, Comm. Math. Phys. 211, no. 3, 527-558 (2000). | MR | Zbl

[MPP2] Morales (C.), Pacifico (M. J.), Pujals (E. R.).— Singular-hyperbolic systems, Proc. Amer. Math. Soc. 127, no. 11, 3393-3401 (1999). | MR | Zbl

[MPu] Morales (C.), Pujals (E. R.).— Singular strange attractors on the boundary of Morse-Smale systems, Ann. Sci. École Norm. Sup. (4) 30, no. 6, 693-717 (1997). | Numdam | MR | Zbl

[PdM] Palis (J.), de Melo (W.).— Geometric theory of dynamical systems. An introduction., Translated from the Portuguese by A. K. Manning. Springer-Verlag, New York- Berlin, 1982. | MR | Zbl

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