# American Institute of Mathematical Sciences

April  2014, 8(2): 191-219. doi: 10.3934/jmd.2014.8.191

## On the singular-hyperbolicity of star flows

 1 School of Mathematical Sciences, Peking University, Beijing 100871, China and Institut de Mathématiques de Bourgogne, Université de Bourgogne, Dijon 21000, France 2 School of Mathematical Sciences, Peking University, Beijing 100871, China 3 School of Mathematic Sciences, Peking University, Beijing, 100871

Received  September 2013 Published  November 2014

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.
Citation: Yi Shi, Shaobo Gan, Lan Wen. On the singular-hyperbolicity of star flows. Journal of Modern Dynamics, 2014, 8 (2) : 191-219. doi: 10.3934/jmd.2014.8.191
##### References:
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show all references

##### References:
 [1] N. Aoki, The set of Axiom A diffeomorphisms with no cycles,, Bol. Soc. Brasil. Mat. (N.S.), 23 (1992), 21. doi: 10.1007/BF02584810. Google Scholar [2] A. Arbieto, C. Morales and B. Santiago, Lyapunov stability and sectional-hyperbolicity for higher-dimensional flows,, Mathematische Annalen, (). doi: 10.1007/s00208-014-1061-3. Google Scholar [3] C. Bonatti and S. Crovisier, Récurrence et généricité,, Invent. Math., 158 (2004), 33. doi: 10.1007/s00222-004-0368-1. Google Scholar [4] 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. Google Scholar [5] C. Bonatti, S. Gan and D. Yang, Dominated chain recurrent classes with singularities,, , (). Google Scholar [6] S. Crovisier, Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms,, Publ. Math. Inst. Hautes Études Sci., 104 (2006), 87. doi: 10.1007/s10240-006-0002-4. Google Scholar [7] J. Franks, Necessary conditions for stability of diffeomorphisms,, Trans. Amer. Math. Soc., 158 (1971), 301. doi: 10.1090/S0002-9947-1971-0283812-3. Google Scholar [8] S. Gan and L. Wen, Heteroclinic cycles and homoclinic closures for generic diffeomorphisms,, J. Dynam. Differential Equations, 15 (2003), 451. doi: 10.1023/B:JODY.0000009743.10365.9d. Google Scholar [9] S. Gan and L. Wen, Nonsingular star flows satisfy Axiom A and the no-cycle condition,, Invent. Math., 164 (2006), 279. doi: 10.1007/s00222-005-0479-3. Google Scholar [10] S. Gan and D. Yang, Morse-Smale systems and horseshoes for three-dimensional singular flows,, , (). Google Scholar [11] J. Guckenheimer, A strange, strange attractor,, in The Hopf Bifurcation Theorems and its Applications, 19 (1976), 368. Google Scholar [12] S. Hayashi, Diffeomorphisms in $\mathcal F^1(M)$ satisfy Axiom A,, Ergod. Th. Dynam. Sys., 12 (1992), 233. doi: 10.1017/S0143385700006726. Google Scholar [13] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137. Google Scholar [14] 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. doi: 10.3934/dcds.2005.13.239. Google Scholar [15] S. Liao, A basic property of a certain class of differential systems,, (in Chinese) Acta Math. Sinica, 22 (1979), 316. Google Scholar [16] S. Liao, Obstruction sets. II,, (in Chinese) Beijing Daxue Xuebao, (1981), 1. Google Scholar [17] 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. Google Scholar [18] S. Liao, On $(\eta,d)$-contractible orbits of vector fields,, Systems Sci. Math. Sci., 2 (1989), 193. Google Scholar [19] R. Mañé, An ergodic closing lemma,, Ann. Math. (2), 116 (1982), 503. doi: 10.2307/2007021. Google Scholar [20] R. Metzger and C. Morales, On sectional-hyperbolic systems,, Ergodic Theory and Dynamical Systems, 28 (2008), 1587. doi: 10.1017/S0143385707000995. Google Scholar [21] 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. Google Scholar [22] C. Morales and M. Pacifico, A dichotomy for three-dimensional vector fields,, Ergodic Theory Dynam. Systems, 23 (2003), 1575. doi: 10.1017/S0143385702001621. Google Scholar [23] J. Palis and S. Smale, Structural stability theorems,, in 1970 Global Analysis (Proc. Sympos. Pure Math., (1968), 223. Google Scholar [24] V. Pliss, A hypothesis due to Smale,, Diff. Eq., 8 (1972), 203. Google Scholar [25] C. Pugh and M. Shub, $\Omega$-stability for flows,, Invent. Math., 11 (1970), 150. doi: 10.1007/BF01404608. Google Scholar [26] C. Pugh and M. Shub, Ergodic elements of ergodic actions,, Compositio Math., 23 (1971), 115. Google Scholar [27] E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms,, Annals of Math. (2), 151 (2000), 961. doi: 10.2307/121127. Google Scholar [28] S. Smale, The $\Omega$-stability theorem,, in 1970 Global Analysis (Proc. Sympos. Pure Math., (1968), 289. Google Scholar [29] L. Wen, On the $C^1$ stability conjecture for flows,, J. Differential Equations, 129 (1996), 334. doi: 10.1006/jdeq.1996.0121. Google Scholar [30] L. Wen and Z. Xia, $C^1$ connecting lemmas,, Trans. Am. Math. Soc., 352 (2000), 5213. doi: 10.1090/S0002-9947-00-02553-8. Google Scholar [31] D. Yang and Y. Zhang, On the finiteness of uniform sinks,, J. Diff. Eq., 257 (2014), 2102. doi: 10.1016/j.jde.2014.05.028. Google Scholar [32] S. Zhu, S. Gan and L. Wen, Indices of singularities of robustly transitive sets,, Discrete Contin. Dyn. Syst., 21 (2008), 945. doi: 10.3934/dcds.2008.21.945. Google Scholar
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