# 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:
 [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. [2] 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. [3] C. Bonatti and S. Crovisier, Récurrence et généricité, Invent. Math., 158 (2004), 33-104. doi: 10.1007/s00222-004-0368-1. [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), 355-418. [5] C. Bonatti, S. Gan and D. Yang, Dominated chain recurrent classes with singularities, arXiv:1106.3905. [6] 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. [7] J. Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc., 158 (1971), 301-308. doi: 10.1090/S0002-9947-1971-0283812-3. [8] 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. [9] 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. [10] S. Gan and D. Yang, Morse-Smale systems and horseshoes for three-dimensional singular flows, arXiv:1302.0946. [11] J. Guckenheimer, A strange, strange attractor, in The Hopf Bifurcation Theorems and its Applications, Applied Mathematical Series, 19, Springer-Verlag, 1976, 368-381. [12] S. Hayashi, Diffeomorphisms in $\mathcal F^1(M)$ satisfy Axiom A, Ergod. Th. Dynam. Sys., 12 (1992), 233-253. doi: 10.1017/S0143385700006726. [13] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173. [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), 239-269. doi: 10.3934/dcds.2005.13.239. [15] S. Liao, A basic property of a certain class of differential systems, (in Chinese) Acta Math. Sinica, 22 (1979), 316-343. [16] S. Liao, Obstruction sets. II, (in Chinese) Beijing Daxue Xuebao, no. 2, (1981), 1-36. [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), 1-19. [18] S. Liao, On $(\eta,d)$-contractible orbits of vector fields, Systems Sci. Math. Sci., 2 (1989), 193-227. [19] R. Mañé, An ergodic closing lemma, Ann. Math. (2), 116 (1982), 503-540. doi: 10.2307/2007021. [20] R. Metzger and C. Morales, On sectional-hyperbolic systems, Ergodic Theory and Dynamical Systems, 28 (2008), 1587-1597. doi: 10.1017/S0143385707000995. [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), 375-432. [22] C. Morales and M. Pacifico, A dichotomy for three-dimensional vector fields, Ergodic Theory Dynam. Systems, 23 (2003), 1575-1600. doi: 10.1017/S0143385702001621. [23] 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. [24] V. Pliss, A hypothesis due to Smale, Diff. Eq., 8 (1972), 203-214. [25] C. Pugh and M. Shub, $\Omega$-stability for flows, Invent. Math., 11 (1970), 150-158. doi: 10.1007/BF01404608. [26] C. Pugh and M. Shub, Ergodic elements of ergodic actions, Compositio Math., 23 (1971), 115-122. [27] E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Annals of Math. (2), 151 (2000), 961-1023. doi: 10.2307/121127. [28] 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. [29] L. Wen, On the $C^1$ stability conjecture for flows, J. Differential Equations, 129 (1996), 334-357. doi: 10.1006/jdeq.1996.0121. [30] 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. [31] 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. [32] 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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##### References:
 [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. [2] 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. [3] C. Bonatti and S. Crovisier, Récurrence et généricité, Invent. Math., 158 (2004), 33-104. doi: 10.1007/s00222-004-0368-1. [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), 355-418. [5] C. Bonatti, S. Gan and D. Yang, Dominated chain recurrent classes with singularities, arXiv:1106.3905. [6] 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. [7] J. Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc., 158 (1971), 301-308. doi: 10.1090/S0002-9947-1971-0283812-3. [8] 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. [9] 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. [10] S. Gan and D. Yang, Morse-Smale systems and horseshoes for three-dimensional singular flows, arXiv:1302.0946. [11] J. Guckenheimer, A strange, strange attractor, in The Hopf Bifurcation Theorems and its Applications, Applied Mathematical Series, 19, Springer-Verlag, 1976, 368-381. [12] S. Hayashi, Diffeomorphisms in $\mathcal F^1(M)$ satisfy Axiom A, Ergod. Th. Dynam. Sys., 12 (1992), 233-253. doi: 10.1017/S0143385700006726. [13] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173. [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), 239-269. doi: 10.3934/dcds.2005.13.239. [15] S. Liao, A basic property of a certain class of differential systems, (in Chinese) Acta Math. Sinica, 22 (1979), 316-343. [16] S. Liao, Obstruction sets. II, (in Chinese) Beijing Daxue Xuebao, no. 2, (1981), 1-36. [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), 1-19. [18] S. Liao, On $(\eta,d)$-contractible orbits of vector fields, Systems Sci. Math. Sci., 2 (1989), 193-227. [19] R. Mañé, An ergodic closing lemma, Ann. Math. (2), 116 (1982), 503-540. doi: 10.2307/2007021. [20] R. Metzger and C. Morales, On sectional-hyperbolic systems, Ergodic Theory and Dynamical Systems, 28 (2008), 1587-1597. doi: 10.1017/S0143385707000995. [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), 375-432. [22] C. Morales and M. Pacifico, A dichotomy for three-dimensional vector fields, Ergodic Theory Dynam. Systems, 23 (2003), 1575-1600. doi: 10.1017/S0143385702001621. [23] 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. [24] V. Pliss, A hypothesis due to Smale, Diff. Eq., 8 (1972), 203-214. [25] C. Pugh and M. Shub, $\Omega$-stability for flows, Invent. Math., 11 (1970), 150-158. doi: 10.1007/BF01404608. [26] C. Pugh and M. Shub, Ergodic elements of ergodic actions, Compositio Math., 23 (1971), 115-122. [27] E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Annals of Math. (2), 151 (2000), 961-1023. doi: 10.2307/121127. [28] 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. [29] L. Wen, On the $C^1$ stability conjecture for flows, J. Differential Equations, 129 (1996), 334-357. doi: 10.1006/jdeq.1996.0121. [30] 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. [31] 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. [32] 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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