Robustly transitive sets and heterodimensional cycles

Bonatti, Christian ; Díaz, Lorenzo J. ; Pujals, Enrique R. ; Rocha, Jorge

Geometric methods in dynamics (I) : Volume in honor of Jacob Palis, Astérisque, no. 286 (2003), 36 p.

MR 2052302 | Zbl 1056.37024 | 1 citation dans Numdam

@incollection{AST_2003__286__187_0,
     author = {Bonatti, Christian and D\'\i az, Lorenzo J. and Pujals, Enrique R. and Rocha, Jorge},
     title = {Robustly transitive sets and heterodimensional cycles},
     booktitle = {Geometric methods in dynamics (I) : Volume in honor of Jacob Palis},
     editor = {de Melo, Wellington and Viana, Marcelo and Yoccoz, Jean-Christophe},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {286},
     year = {2003},
     zbl = {1056.37024},
     mrnumber = {2052302},
     language = {en},
     url = {www.numdam.org/item/AST_2003__286__187_0/}
}

Bonatti, Christian; Díaz, Lorenzo J.; Pujals, Enrique R.; Rocha, Jorge. Robustly transitive sets and heterodimensional cycles, dans Geometric methods in dynamics (I) : Volume in honor of Jacob Palis, Astérisque, no. 286 (2003), 36 p. http://www.numdam.org/item/AST_2003__286__187_0/

Bibliographie

[Ar] M.-C. Arnaud, Creation de connexions en topologie $C^{1}$ , Ergod. Th. & Dynam. Syst., 21, 339-381, (2001). | MR 1827109 | Zbl 0997.37007

[AS] R. Abraham and S. Smale, Nongenericity of $Ω$ -stability, Global Analysis I, Proc. Symp. Pure Math. A.M.S., 14, 5-8, (1968). | Article | MR 271986 | Zbl 0215.25102

[B] Ch. Bonatti, A local mechanism for robust transitivity, Lecture at the Seminar of Dynamical Systems, IMPA, August (1996).

[BD1] Ch. Bonatti and L. J. Díaz, Persistence of transitive diffeomorphims, Ann. Math., 143, 367-396, (1995).

[BDP] Ch. Bonatti, L. J. Díaz, and E. R. Pujals, A $C^{1}$ -generic dichotomy for diffeomorphisms: weak forms of hyperbolicicity or infinitely many sinks or sources, preprint PUC-Rio (1999) and to appear in Annals of Math.. | MR 2018925 | Zbl 1049.37011

[BV] Ch. Bonatti and M. Viana, $S R B$ for partially hyperbolic attractors: the contracting case, Israel Journal of Math., 115, 157-193, (2000). | MR 1749677 | Zbl 0996.37033

[CMP] C. Carballo, C. Morales, and M. J. Pacífico, Homoclinic classes for $C^{1}$ -generic vector fields, preprint PUC-Rio (2000) and to appear in Ergod. Th. & Dynam. Syst. | MR 1972228 | Zbl 1047.37009

[D] L. J. Díaz, Robust nonhyperbolic dynamics and heterodimensional cycles, Ergod. Th. & Dynam. Syst., 15, 291-315, (1995). | MR 1332405 | Zbl 0831.58035

[DR] L. J. Díaz and J. Rocha, Partially hyperbolic and transitive dynamics generated by heterochnic cycles, Ergod. Th. & Dynam. Syst., 25, 25-76, (2001). | MR 1826660 | Zbl 0972.37018

[DPU] L. J. Díaz, E. R. Pujals, and R. Ures, Partial hyperbolicity and robust transitivity, Acta Mathematica, 183, 1-43, (1999). | Article | MR 1719547 | Zbl 0987.37020

[F] J. Franks, Necessary conditions for stability of diffeomorphisms, Trans. A.M.S., 158, 301-308, (1971). | Article | MR 283812 | Zbl 0219.58005

[H] S. Hayashi, Connecting invariant manifolds and the solution of the $C^{1}$ -stability and $Ω$ -stability conjectures for flows, Ann. of Math., 145, 81-137, (1997). | Article | MR 1432037 | Zbl 0871.58067

[M1] R. Mañé, Contributions to the stability conjecture, Topology, 17, 386-396, (1978). | Article | MR 516217 | Zbl 0405.58035

[M2] R. Mañé, An ergodic closing lemma, Ann. of Math., 116, 503-540, (1982). | Article | MR 678479 | Zbl 0511.58029

[N] S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13, 9-18, (1974). | Article | MR 339291 | Zbl 0275.58016

[R] N. Romero, Persistence of homoclinic tangencies in higher dimensions, Ergod. Th. & Dynam. Syst., 15, 735-759, (1995). | Article | MR 1346398 | Zbl 0833.58020

[Pa] J. Palis, A global view of dynamics and a conjecture on the denseness and finitude of attractors, Astérisque, 261, 335-347, (2000). | Numdam | MR 1755446 | Zbl 1044.37014

[PV] J. Palis and M. Viana, High dimension diffeomorphisms displaying infinitely many sinks, Ann. of Math., 140, 207-250, (1994). | Article | MR 1289496 | Zbl 0817.58004

[Pu] C. Pugh, The closing lemma, Amer. Jour. of Math., 89, 956-1009, (1967). | Article | MR 226669 | Zbl 0167.21803

[Sh] M. Shub, Topological transitive diffeomorphism on $T^{4}$ , Lect. Notes in Math., 206, 39 (1971).

[Si] R. C. Simon, A $3$ -dimensional Abraham-Smale example, Proc. A.M.S., 34(2), 629-630, (1972). | MR 295391 | Zbl 0259.58006

[Sm] S. Smale, Differentiable dynamical systems, Bull. A.M.S., 73, 747-817, (1967). | Article | MR 228014 | Zbl 0202.55202

[YA] J. Yorke and K. Alligood, Cascades of period doubling bifurcations: a prerequisite for horseshoes, Bull. Amer. Math. Soc. (New Series), 9, 319-323, (1983). | Article | MR 714994 | Zbl 0541.58039