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.

[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.

[3]

C. Bonatti and S. Crovisier, Récurrence et généricité,, Invent. Math., 158 (2004), 33. 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.

[5]

C. Bonatti, S. Gan and D. Yang, Dominated chain recurrent classes with singularities,, , ().

[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.

[7]

J. Franks, Necessary conditions for stability of diffeomorphisms,, Trans. Amer. Math. Soc., 158 (1971), 301. 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. 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. doi: 10.1007/s00222-005-0479-3.

[10]

S. Gan and D. Yang, Morse-Smale systems and horseshoes for three-dimensional singular flows,, , ().

[11]

J. Guckenheimer, A strange, strange attractor,, in The Hopf Bifurcation Theorems and its Applications, 19 (1976), 368.

[12]

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

[13]

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

[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.

[15]

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

[16]

S. Liao, Obstruction sets. II,, (in Chinese) Beijing Daxue Xuebao, (1981), 1.

[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.

[18]

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

[19]

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

[20]

R. Metzger and C. Morales, On sectional-hyperbolic systems,, Ergodic Theory and Dynamical Systems, 28 (2008), 1587. 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.

[22]

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

[23]

J. Palis and S. Smale, Structural stability theorems,, in 1970 Global Analysis (Proc. Sympos. Pure Math., (1968), 223.

[24]

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

[25]

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

[26]

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

[27]

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

[28]

S. Smale, The $\Omega$-stability theorem,, in 1970 Global Analysis (Proc. Sympos. Pure Math., (1968), 289.

[29]

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

[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.

[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.

[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.

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.

[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.

[3]

C. Bonatti and S. Crovisier, Récurrence et généricité,, Invent. Math., 158 (2004), 33. 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.

[5]

C. Bonatti, S. Gan and D. Yang, Dominated chain recurrent classes with singularities,, , ().

[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.

[7]

J. Franks, Necessary conditions for stability of diffeomorphisms,, Trans. Amer. Math. Soc., 158 (1971), 301. 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. 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. doi: 10.1007/s00222-005-0479-3.

[10]

S. Gan and D. Yang, Morse-Smale systems and horseshoes for three-dimensional singular flows,, , ().

[11]

J. Guckenheimer, A strange, strange attractor,, in The Hopf Bifurcation Theorems and its Applications, 19 (1976), 368.

[12]

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

[13]

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

[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.

[15]

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

[16]

S. Liao, Obstruction sets. II,, (in Chinese) Beijing Daxue Xuebao, (1981), 1.

[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.

[18]

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

[19]

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

[20]

R. Metzger and C. Morales, On sectional-hyperbolic systems,, Ergodic Theory and Dynamical Systems, 28 (2008), 1587. 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.

[22]

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

[23]

J. Palis and S. Smale, Structural stability theorems,, in 1970 Global Analysis (Proc. Sympos. Pure Math., (1968), 223.

[24]

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

[25]

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

[26]

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

[27]

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

[28]

S. Smale, The $\Omega$-stability theorem,, in 1970 Global Analysis (Proc. Sympos. Pure Math., (1968), 289.

[29]

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

[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.

[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.

[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.

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