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

$C^1$ weak Palis conjecture for nonsingular flows

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.

Citation: Qianying Xiao, Zuohuan Zheng. $C^1$ weak Palis conjecture for nonsingular flows. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1809-1832. doi: 10.3934/dcds.2018074
References:
[1]

F. AbdenurC. 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.

[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.

[3]

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

[4]

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

[5]

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

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[12]

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

[13]

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

[14]

M. LiS. 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.

[15]

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

[16]

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

[17]

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

[18]

C. MoralesM. 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.

[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.

[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.

[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.

[22]

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

[23]

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

[24]

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

[25]

Y. ShiS. 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.

[26]

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

[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.

[28]

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

[29]

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

[30]

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

[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.

[32]

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

[33]

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

[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.

[35]

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

[36]

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

[37]

S. ZhuS. 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.

show all references

References:
[1]

F. AbdenurC. 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.

[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.

[3]

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

[4]

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

[5]

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

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[12]

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

[13]

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

[14]

M. LiS. 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.

[15]

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

[16]

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

[17]

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

[18]

C. MoralesM. 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.

[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.

[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.

[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.

[22]

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

[23]

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

[24]

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

[25]

Y. ShiS. 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.

[26]

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

[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.

[28]

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

[29]

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

[30]

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

[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.

[32]

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

[33]

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

[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.

[35]

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

[36]

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

[37]

S. ZhuS. 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.

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