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. 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

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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