# zbMATH — the first resource for mathematics

Partial hyperbolicity and robust transitivity. (English) Zbl 0987.37020
Let $$M$$ denote a three-dimensional boundaryless compact manifold and $$\text{Diff}(M)$$ the space of $$C^1$$-diffeomorphism defined on $$M$$ endowed with the usual $$C^1$$-topology. The maximal invariant set of $$\varphi$$ in an open set $$U$$, denoted by $$\Lambda_\varphi(U)$$, is the set of points whose whole orbit is contained in $$U$$. This paper is devoted to the hyperbolicity (uniform partial and strong partial) of $$\Lambda_4(U)$$ derived from its robust transitivity. In the case $$U=M$$, $$\varphi$$ is robustly transitive.

##### MSC:
 37D30 Partially hyperbolic systems and dominated splittings 37C20 Generic properties, structural stability of dynamical systems
##### Keywords:
compact manifold; robust transitivity
Full Text:
##### References:
 [1] Bonatti, Ch., Seminar, IMPA, 1996. [2] Bonatti, Ch. &Díaz, L. J., Persistence of transitive diffeomorphisms.Ann. of Math., 143 (1995), 367–396. [3] –, Connexions hétérocliniques et généricité d’une infinité de puits ou de sources.Ann. Sci. École Norm. Sup., 32 (1999), 135–150. [4] Bonatti, Ch. & Viana, M., SRB measures for partially hyperbolic atractors: the contracting case. To appear inIsrael J. Math. [5] Carvalho, M., Sinai-Ruelle-Bowen measures forN-dimensional derived from Anosov diffeomorphisms.Ergodic Theory Dynamical Systems, 13 (1993), 21–44. · Zbl 0781.58030 [6] Camacho, C. &Lins Neto, A.,Geometric Theory of Foliations, Birkhäuser Boston, Boston, MA, 1985. · Zbl 0568.57002 [7] Díaz, L. J., Robust nonhyperbolic dynamics at heterodimensional cycles.Ergodic Theory Dynamical Systems, 15 (1995), 291–315. · Zbl 0831.58035 [8] –, Persistence of cycles and nonhyperbolic dynamics at heteroclinic bifurcations.Nonlinearity, 8 (1995), 693–715. · Zbl 0836.58031 [9] Doering, C. I., Persistently transitive vector fields on three-dimensional manifolds, inDynamical Systems and Bifurcation Theory (Rio de Janeiro, 1985), pp. 59–89. Pitman Res. Notes Math. Ser., 160, Longman Sci. Tech., Harlow, 1987. [10] Díaz, L. J. &Rocha, J., Noncritical saddle-node cycles and robust nonhyperbolic dynamics.Dynamics Stability Systems, 12 (1997), 109–135. · Zbl 0883.58023 [11] Díaz, L. J. &Ures, R., Persistent homoclinic tangencies and the unfolding of cycles.Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 643–659. [12] Franks, J., Necessary conditions for the stability of diffeomorphisms.Trans. Amer. Math. Soc., 158 (1971), 301–308. · Zbl 0219.58005 [13] Grayson, M., Pugh, C. &Shub, M., Stably ergodic diffeomorphisms.Ann. of Math., 140 (1994), 295–329. · Zbl 0824.58032 [14] Hayashi, S., Connecting invariant manifolds and the solution of theC 1-stability and {$$\Omega$$}-stability conjectures for flows.Ann. of Math., 145 (1997), 81–137. · Zbl 0871.58067 [15] Hirsch, M., Pugh, C. &Shub, M.,Invariant Manifolds, Lecture Notes in Math., 583. Springer-Verlag, Berlin-New York, 1977. · Zbl 0355.58009 [16] Mañé, R., Contributions to the stability conjecture.Topology, 17 (1978), 386–396. · Zbl 0405.58035 [17] –, Persistent manifolds are normally hyperbolic.Trans. Amer. Math. Soc., 246 (1978). 261–283. · Zbl 0362.58014 [18] –, An ergodic closing lemma.Ann. of Math., 116 (1982), 541–558. · Zbl 0519.58003 [19] –, A proof of theC 1 stability conjecture.Inst. Hautes Études Sci. Publ. Math., 66 (1988), 161–210. · Zbl 0678.58022 [20] Morales, C., Pacífico, M. J. &Pujals, E. R., OnC 1 robust singular transitive sets for three-dimensional flows.C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 81–86. · Zbl 0918.58036 [21] Newhouse, S., Codimension one Anosov diffeomorphisms.Amer. J. Math., 92 (1970), 761–770. · Zbl 0204.56901 [22] Pugh, C., The closing lemma.Amer. J. Math., 89 (1967), 956–1009. · Zbl 0167.21803 [23] Palis, J. &Viana, M., High-dimensional diffeomorphisms displaying infinitely many sinks.Ann. of Math., 140 (1994), 207–250. · Zbl 0817.58004 [24] Romero, N., Persistence of homoclinic tangencies in higher dimension.Ergodic Theory Dynamical Systems, 15 (1995), 735–759. · Zbl 0833.58020 [25] Shub, M., Topologically transitive diffeomorphism ofT 4, inSymposium on Differential Equations and Dynamical Systems (University of Warwick, 1968/69), pp. 39–40. Lecture Notes in Math., 206. Springer-Verlag, Berlin-New York, 1971. [26] Smale, S., Differentiable dynamical systems.Bull. Amer. Math. Soc., 73 (1967), 147–817. · Zbl 0202.55202 [27] Williams, R. F., The ”DA” maps of Smale and structural stability, inGlobal Analysis (Berkeley, CA, 1968), pp. 329–334. Proc. Sympos. Pure Math., 14. Amer. Math. Soc., Providence, RI, 1970.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.