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April  2018, 38(4): 1809-1832. doi: 10.3934/dcds.2018074

$C^1$ weak Palis conjecture for nonsingular flows

Qianying Xiao 1,, and  Zuohuan Zheng 2,3,

1. 

School of Mathematical Sciences, Peking University, Beijing 100871, China
2. 

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
3. 

University of Chinese Academy of Sciences, Beijing 100049, China

* Corresponding author: Qianying Xiao

Received  September 2016 Revised  October 2017 Published  January 2018

This paper focuses on generic properties of continuous dynamical systems. We prove $C^1$ weak Palis conjecture for nonsingular flows: Morse-Smale vector fields and vector fields admitting horseshoes are open and dense among $C^1$ nonsingular vector fields.

Our arguments contain three main ingredients: linear Poincaré flow, Liao's selecting lemma and the adapting of Crovisier's central model.

Firstly, by studying the linear Poincaré flow, we prove for a $C^1$ generic vector field away from horseshoes, any non-trivial nonsingular chain recurrent class contains a minimal set which is partially hyperbolic with 1-dimensional center with respect to the linear Poincaré flow.

Secondly, to understand the neutral behaviour of the 1-dimensional center, we adapt Crovisier's central model. The difficulties are that we can not build invariant plaque family of any time, the periodic point of a flow is not periodic for the discrete time map. Through delicate analysis of the center manifold of a periodic orbit near the partially hyperbolic set, we manage to yield nice periodic points such that their stable manifolds and unstable manifolds are well-placed for transverse intersection.

Keywords: Nonsingular flow, Morse-Smale system, away from horseshoe, linear Poincaré flow, central model.
Mathematics Subject Classification: Primary: 37C20, 37C29; Secondary: 37D15.
Citation: Qianying Xiao, Zuohuan Zheng. $C^1$ weak Palis conjecture for nonsingular flows. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 1809-1832. doi: 10.3934/dcds.2018074
References:
[1]

F. Abdenur, C. Bonatti and S. Crovisier, Nonuniform hyperbolicity for $C^1$-generic diffeomorphisms, Israel J. Math., 183 (2011), 1-60.  doi: 10.1007/s11856-011-0041-5.  Google Scholar

[2]

A. Arroyo and F. Rodriguez Hertz, Homoclinic bifurcations and uniform hyperbolicity for three dimensional flows, Ann. I. H. Poincaré-AN, 20 (2003), 805-841.  doi: 10.1016/S0294-1449(03)00016-7.  Google Scholar

[3]

C. Bonatti and S. Crovisier, Récurrence et généricité, Invent. Math., 158 (2004), 33-104.   Google Scholar

[4]

C. Bonatti, S. Gan and L. Wen, On the existence of non-trivial homoclinic classes, Ergodic Theory Dynam. Systems, 27 (2007), 1473-1508.   Google Scholar

[5]

C. Bonatti, S. Gan and D. Yang, Dominated chain recurrent classes with singularities, Ann. Sc. Norm. Super. Pisa Cl. Sci.(5), 14 (2015), 83-99.   Google Scholar

[6]

C. Conley, Isolated Invariant Sets and the Morse Index, Regional Conference Series in Mathematics, 38. American Mathematical Society, Providence, R. I., 1978. doi: 0-8218-1688-8.  Google Scholar

[7]

S. Crovisier, Periodic orbits and chain transitive sets of $C^1$-diffeomorphisms, Publ. Math. Inst. Hautes Études Sci., 104 (2006), 87-141.   Google Scholar

[8]

S. Crovisier, Birth of homoclinic intersections: A model for the central dynamics of partially hyperbolic systems, Ann. Math., 172 (2010), 1641-1677.  doi: 10.4007/annals.2010.172.1641.  Google Scholar

[9]

J. Franks and J. Selgrade, Hyperbolicity and chain recurrence, J. Diff. Equa., 26 (1977), 27-36.  doi: 10.1016/0022-0396(77)90096-1.  Google Scholar

[10]

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.  Google Scholar

[11]

S. Gan and D. Yang, Morse-Smale systems and horseshoes for three dimensional singular flows, J. Eur. Math. Soc., to appear. Google Scholar

[12]

J. Guchenheimer and R. Williams, Structural stability of Lorenz attractors, Publ. Math. Inst. Hautes Études Sci., 50 (1979), 59-72.   Google Scholar

[13]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin-New York, 1977. Google Scholar

[14]

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

[15]

S. Liao, Qualitative Theory of Differentiable Dyamical Systems, China Science Press, Beijing, 1996. doi: 9787030054371.  Google Scholar

[16]

S. Liao, Obstruction sets(Ⅰ), Acta Math. Sinica, 23 (1980), 411-453.   Google Scholar

[17]

S. Liao, On $(η,d)$ -contractible orbits of vector fields, Systems Science and Math. Sciences, 2 (1989), 193-227.   Google Scholar

[18]

C. Morales, M. Pacifico and E. Pujals, Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers, Ann. Math., 160 (2004), 375-432.  doi: 10.4007/annals.2004.160.375.  Google Scholar

[19]

J. Palis, A global view of dynamics and a conjecture on the denseness of finitude of attractors, Géométrie complexe et systemes dynamiques, Astérisque, 261 (2000), 335-347.   Google Scholar

[20]

J. Palis, A global perspective for non-conservative dynamics, Ann. I. H. Poincaré, 22 (2005), 485-507.  doi: 10.1016/j.anihpc.2005.01.001.  Google Scholar

[21]

J. Palis and W. de Melo, Geometric Theory of Dynamical Systems, An introduction, Translated from the Portuguese by A. K. Manning, Springer-Verlag, New York-Berlin, 1982. doi: 0-387-90668-1.  Google Scholar

[22]

J. Palis and S. Smale, Structural stability theorems, Amer. Math. Soc. , Providence, R. I. 1970 Global Analysis, 223–231.  Google Scholar

[23]

M. Peixoto, Structural stability on two dimensional manifolds, Topology, 1 (1962), 101-120.  doi: 10.1016/0040-9383(65)90018-2.  Google Scholar

[24]

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

[25]

Y. Shi, S. Gan and L. Wen, On the singular hyperbolicity of star flows, J. Mod. Dyn., 8 (2014), 191-219.  doi: 10.3934/jmd.2014.8.191.  Google Scholar

[26]

S. Smale, On gradient dynamical systems, Ann. Math.(2), 74 (1961), 199-206.  doi: 10.2307/1970311.  Google Scholar

[27]

S. Smale, Diffeomorphisms with many periodic points, in 1965 Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N. J., (1965), 63–80.  Google Scholar

[28]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

[29]

S. Smale, The Mathematics of Time, Essays on dynamical systems, economic processes, and related topics, Springer-Verlag, New York-Heidelberg-Berlin, 1980.  Google Scholar

[30]

L. Wen, Homoclinic tangencies and dominated splittings, Nonlinearity, 15 (2002), 1445-1469.  doi: 10.1088/0951-7715/15/5/306.  Google Scholar

[31]

L. Wen, Generic diffeomorphisms away from homoclinic tangencies and heterodimensional cycles, Bull. Braz. Math. Soc.(N.S.), 35 (2004), 419-452.  doi: 10.1007/s00574-004-0023-x.  Google Scholar

[32]

L. Wen, The selecting lemma of Liao, Disc. Cont. Dynam. Syst., 20 (2008), 159-175.   Google Scholar

[33]

L. Wen, Differential Dyamical Systems, Higher Education Press, 2015(in Chinese). Google Scholar

[34]

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

[35]

Q. Xiao, Estimates of products of norms for flows away from homoclinic tangency. arXiv: 1705.04072. Google Scholar

[36]

R. Zheng, Partially Hyperbolicity of Vector Fields Away from Horseshoe, Ph. D thesis, Peking University, 2015. Google Scholar

[37]

S. Zhu, S. Gan and L. Wen, Indices of singularities of robustly transitive sets, Disc. Cont. Dynam. Syst., 21 (2008), 945-957.  doi: 10.3934/dcds.2008.21.945.  Google Scholar

show all references

References:
[1]

F. Abdenur, C. Bonatti and S. Crovisier, Nonuniform hyperbolicity for $C^1$-generic diffeomorphisms, Israel J. Math., 183 (2011), 1-60.  doi: 10.1007/s11856-011-0041-5.  Google Scholar

[2]

A. Arroyo and F. Rodriguez Hertz, Homoclinic bifurcations and uniform hyperbolicity for three dimensional flows, Ann. I. H. Poincaré-AN, 20 (2003), 805-841.  doi: 10.1016/S0294-1449(03)00016-7.  Google Scholar

[3]

C. Bonatti and S. Crovisier, Récurrence et généricité, Invent. Math., 158 (2004), 33-104.   Google Scholar

[4]

C. Bonatti, S. Gan and L. Wen, On the existence of non-trivial homoclinic classes, Ergodic Theory Dynam. Systems, 27 (2007), 1473-1508.   Google Scholar

[5]

C. Bonatti, S. Gan and D. Yang, Dominated chain recurrent classes with singularities, Ann. Sc. Norm. Super. Pisa Cl. Sci.(5), 14 (2015), 83-99.   Google Scholar

[6]

C. Conley, Isolated Invariant Sets and the Morse Index, Regional Conference Series in Mathematics, 38. American Mathematical Society, Providence, R. I., 1978. doi: 0-8218-1688-8.  Google Scholar

[7]

S. Crovisier, Periodic orbits and chain transitive sets of $C^1$-diffeomorphisms, Publ. Math. Inst. Hautes Études Sci., 104 (2006), 87-141.   Google Scholar

[8]

S. Crovisier, Birth of homoclinic intersections: A model for the central dynamics of partially hyperbolic systems, Ann. Math., 172 (2010), 1641-1677.  doi: 10.4007/annals.2010.172.1641.  Google Scholar

[9]

J. Franks and J. Selgrade, Hyperbolicity and chain recurrence, J. Diff. Equa., 26 (1977), 27-36.  doi: 10.1016/0022-0396(77)90096-1.  Google Scholar

[10]

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.  Google Scholar

[11]

S. Gan and D. Yang, Morse-Smale systems and horseshoes for three dimensional singular flows, J. Eur. Math. Soc., to appear. Google Scholar

[12]

J. Guchenheimer and R. Williams, Structural stability of Lorenz attractors, Publ. Math. Inst. Hautes Études Sci., 50 (1979), 59-72.   Google Scholar

[13]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin-New York, 1977. Google Scholar

[14]

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

[15]

S. Liao, Qualitative Theory of Differentiable Dyamical Systems, China Science Press, Beijing, 1996. doi: 9787030054371.  Google Scholar

[16]

S. Liao, Obstruction sets(Ⅰ), Acta Math. Sinica, 23 (1980), 411-453.   Google Scholar

[17]

S. Liao, On $(η,d)$ -contractible orbits of vector fields, Systems Science and Math. Sciences, 2 (1989), 193-227.   Google Scholar

[18]

C. Morales, M. Pacifico and E. Pujals, Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers, Ann. Math., 160 (2004), 375-432.  doi: 10.4007/annals.2004.160.375.  Google Scholar

[19]

J. Palis, A global view of dynamics and a conjecture on the denseness of finitude of attractors, Géométrie complexe et systemes dynamiques, Astérisque, 261 (2000), 335-347.   Google Scholar

[20]

J. Palis, A global perspective for non-conservative dynamics, Ann. I. H. Poincaré, 22 (2005), 485-507.  doi: 10.1016/j.anihpc.2005.01.001.  Google Scholar

[21]

J. Palis and W. de Melo, Geometric Theory of Dynamical Systems, An introduction, Translated from the Portuguese by A. K. Manning, Springer-Verlag, New York-Berlin, 1982. doi: 0-387-90668-1.  Google Scholar

[22]

J. Palis and S. Smale, Structural stability theorems, Amer. Math. Soc. , Providence, R. I. 1970 Global Analysis, 223–231.  Google Scholar

[23]

M. Peixoto, Structural stability on two dimensional manifolds, Topology, 1 (1962), 101-120.  doi: 10.1016/0040-9383(65)90018-2.  Google Scholar

[24]

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

[25]

Y. Shi, S. Gan and L. Wen, On the singular hyperbolicity of star flows, J. Mod. Dyn., 8 (2014), 191-219.  doi: 10.3934/jmd.2014.8.191.  Google Scholar

[26]

S. Smale, On gradient dynamical systems, Ann. Math.(2), 74 (1961), 199-206.  doi: 10.2307/1970311.  Google Scholar

[27]

S. Smale, Diffeomorphisms with many periodic points, in 1965 Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N. J., (1965), 63–80.  Google Scholar

[28]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

[29]

S. Smale, The Mathematics of Time, Essays on dynamical systems, economic processes, and related topics, Springer-Verlag, New York-Heidelberg-Berlin, 1980.  Google Scholar

[30]

L. Wen, Homoclinic tangencies and dominated splittings, Nonlinearity, 15 (2002), 1445-1469.  doi: 10.1088/0951-7715/15/5/306.  Google Scholar

[31]

L. Wen, Generic diffeomorphisms away from homoclinic tangencies and heterodimensional cycles, Bull. Braz. Math. Soc.(N.S.), 35 (2004), 419-452.  doi: 10.1007/s00574-004-0023-x.  Google Scholar

[32]

L. Wen, The selecting lemma of Liao, Disc. Cont. Dynam. Syst., 20 (2008), 159-175.   Google Scholar

[33]

L. Wen, Differential Dyamical Systems, Higher Education Press, 2015(in Chinese). Google Scholar

[34]

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

[35]

Q. Xiao, Estimates of products of norms for flows away from homoclinic tangency. arXiv: 1705.04072. Google Scholar

[36]

R. Zheng, Partially Hyperbolicity of Vector Fields Away from Horseshoe, Ph. D thesis, Peking University, 2015. Google Scholar

[37]

S. Zhu, S. Gan and L. Wen, Indices of singularities of robustly transitive sets, Disc. Cont. Dynam. Syst., 21 (2008), 945-957.  doi: 10.3934/dcds.2008.21.945.  Google Scholar

Figure 1.  Central segment and heteroclinic cycle
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Figure 2.  the local stable manifold of $\gamma_{\hat{x}}$
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Figure 3.  $z$ and $P_{\hat x}^k$ in a twisted position
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