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

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.

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.

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DOI : https://doi.org/10.5802/afst.1180
@article{AFST_2008_6_17_1_193_0,
     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, Toulouse},
     volume = {Ser. 6, 17},
     number = {1},
     year = {2008},
     doi = {10.5802/afst.1180},
     zbl = {pre05380233},
     mrnumber = {2464098},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1180/}
}
C. A Morales. Poincaré-Hopf index and partial hyperbolicity. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 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 656286 | Zbl 0506.58023

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

[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 2105774 | Zbl 1060.37020

[BM] Bing (R. H.), Martin (J. M.), Cubes with knotted holes, Trans. Amer. Math. Soc. 155, 217-231 (1971). | MR 278287 | Zbl 0213.25005

[BMo] Bautista (S.), Morales (C.).— Existence of periodic orbits for singular-hyperbolic sets, Mosc. Math. J. 6, no. 2, 265-297 (2006). | MR 2270614 | Zbl 1124.37021

[Br] Brunella (M.).— Separating basic sets of a nontransitive Anosov flow, Bull. London Math. Soc. 25, 487-490 (1993). | MR 1233413 | Zbl 0790.58028

[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 511133 | Zbl 0397.34056

[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 1489204 | Zbl 0894.55002

[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 1303278 | Zbl 0811.34018

[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 934253 | Zbl 0664.53017

[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 1325916 | Zbl 0862.58006

[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 1810537 | Zbl 0981.57008

[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 2020377 | Zbl 1077.37025

[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 2096543

[GW] Guckenheimer (J.), Williams (R.).— Structural stability of Lorenz attractors, Publ Math IHES 50, 59-72 (1979). | Numdam | MR 556582 | Zbl 0436.58018

[HK] Hasselblatt (B.), Katov (A.).— Introduction to the modern theory of dynamical systems, Cambridge University Press, Cambridge (1995). | MR 1326374 | Zbl 0878.58020

[HPS] Hirsch (M.), Pugh (C.), Shub (M.).— Invariant manifolds, Lec. Not. in Math. 583 (1977), Springer-Verlag. | MR 501173 | Zbl 0355.58009

[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 0976.54046

[M1] Morales (C.).— Examples of singular-hyperbolic attracting sets, Dyn. Syst. (To appear). | MR 2354969 | Zbl 1153.37016

[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 0974.37040

[Mi] Milnor (J.).— Topology from the differentiable viewpoint, Based on notes by David W. Weaver The University Press of Virginia, Charlottesville, Va. 1965. | MR 226651 | Zbl 0136.20402

[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 1773807 | Zbl 0957.37032

[MPP2] Morales (C.), Pacifico (M. J.), Pujals (E. R.).— Singular-hyperbolic systems, Proc. Amer. Math. Soc. 127, no. 11, 3393-3401 (1999). | MR 1610761 | Zbl 0924.58068

[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 1476293 | Zbl 0911.58022

[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 669541 | Zbl 0491.58001