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On the singular-hyperbolicity of star flows

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  • We prove for a generic star vector field $X$ that if, for every chain recurrent class $C$ of $X$, all singularities in $C$ have the same index, then the chain recurrent set of $X$ is singular-hyperbolic. We also prove that every Lyapunov stable chain recurrent class of a generic star vector field is singular-hyperbolic. As a corollary, we prove that the chain recurrent set of a generic 4-dimensional star flow is singular-hyperbolic.
    Mathematics Subject Classification: Primary: 37D30; Secondary: 37D50.


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  • [1]

    N. Aoki, The set of Axiom A diffeomorphisms with no cycles, Bol. Soc. Brasil. Mat. (N.S.), 23 (1992), 21-65.doi: 10.1007/BF02584810.


    A. Arbieto, C. Morales and B. Santiago, Lyapunov stability and sectional-hyperbolicity for higher-dimensional flows, Mathematische Annalen, arXiv:1201.1464. doi: 10.1007/s00208-014-1061-3.


    C. Bonatti and S. Crovisier, Récurrence et généricité, Invent. Math., 158 (2004), 33-104.doi: 10.1007/s00222-004-0368-1.


    C. Bonatti, L. J. Díaz and E. R. Pujals, A $C^1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, Annals of Math. (2), 158 (2003), 355-418.


    C. Bonatti, S. Gan and D. Yang, Dominated chain recurrent classes with singularities, arXiv:1106.3905.


    S. Crovisier, Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms, Publ. Math. Inst. Hautes Études Sci., 104 (2006), 87-141.doi: 10.1007/s10240-006-0002-4.


    J. Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc., 158 (1971), 301-308.doi: 10.1090/S0002-9947-1971-0283812-3.


    S. Gan and L. Wen, Heteroclinic cycles and homoclinic closures for generic diffeomorphisms, J. Dynam. Differential Equations, 15 (2003), 451-471.doi: 10.1023/B:JODY.0000009743.10365.9d.


    S. Gan and L. Wen, Nonsingular star flows satisfy Axiom A and the no-cycle condition, Invent. Math., 164 (2006), 279-315.doi: 10.1007/s00222-005-0479-3.


    S. Gan and D. Yang, Morse-Smale systems and horseshoes for three-dimensional singular flows, arXiv:1302.0946.


    J. Guckenheimer, A strange, strange attractor, in The Hopf Bifurcation Theorems and its Applications, Applied Mathematical Series, 19, Springer-Verlag, 1976, 368-381.


    S. Hayashi, Diffeomorphisms in $\mathcal F^1(M)$ satisfy Axiom A, Ergod. Th. Dynam. Sys., 12 (1992), 233-253.doi: 10.1017/S0143385700006726.


    A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173.


    M. Li, S. Gan and L. Wen, Robustly transitive singular sets via approach of extended linear Poincaré flow, Discrete Contin. Dyn. Syst., 13 (2005), 239-269.doi: 10.3934/dcds.2005.13.239.


    S. Liao, A basic property of a certain class of differential systems, (in Chinese) Acta Math. Sinica, 22 (1979), 316-343.


    S. Liao, Obstruction sets. II, (in Chinese) Beijing Daxue Xuebao, no. 2, (1981), 1-36.


    S. Liao, Certain uniformity properties of differential systems and a generalization of an existence theorem for periodic orbits, (in Chinese) Acta Sci. Natur. Univ. Pekinensis, 2 (1981), 1-19.


    S. Liao, On $(\eta,d)$-contractible orbits of vector fields, Systems Sci. Math. Sci., 2 (1989), 193-227.


    R. Mañé, An ergodic closing lemma, Ann. Math. (2), 116 (1982), 503-540.doi: 10.2307/2007021.


    R. Metzger and C. Morales, On sectional-hyperbolic systems, Ergodic Theory and Dynamical Systems, 28 (2008), 1587-1597.doi: 10.1017/S0143385707000995.


    C. Morales, M. Pacifico and E. Pujals, Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers, Ann. Math. (2), 160 (2004), 375-432.


    C. Morales and M. Pacifico, A dichotomy for three-dimensional vector fields, Ergodic Theory Dynam. Systems, 23 (2003), 1575-1600.doi: 10.1017/S0143385702001621.


    J. Palis and S. Smale, Structural stability theorems, in 1970 Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I, 1970, 223-231.


    V. Pliss, A hypothesis due to Smale, Diff. Eq., 8 (1972), 203-214.


    C. Pugh and M. Shub, $\Omega$-stability for flows, Invent. Math., 11 (1970), 150-158.doi: 10.1007/BF01404608.


    C. Pugh and M. Shub, Ergodic elements of ergodic actions, Compositio Math., 23 (1971), 115-122.


    E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Annals of Math. (2), 151 (2000), 961-1023.doi: 10.2307/121127.


    S. Smale, The $\Omega$-stability theorem, in 1970 Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, 289-297.


    L. Wen, On the $C^1$ stability conjecture for flows, J. Differential Equations, 129 (1996), 334-357.doi: 10.1006/jdeq.1996.0121.


    L. Wen and Z. Xia, $C^1$ connecting lemmas, Trans. Am. Math. Soc., 352 (2000), 5213-5230.doi: 10.1090/S0002-9947-00-02553-8.


    D. Yang and Y. Zhang, On the finiteness of uniform sinks, J. Diff. Eq., 257 (2014), 2102-2114.doi: 10.1016/j.jde.2014.05.028.


    S. Zhu, S. Gan and L. Wen, Indices of singularities of robustly transitive sets, Discrete Contin. Dyn. Syst., 21 (2008), 945-957.doi: 10.3934/dcds.2008.21.945.

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