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April
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 |
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 and 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. |
| [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. Bonatti, S. Gan and L. Wen,
On the existence of non-trivial homoclinic classes, Ergodic Theory Dynam. Systems, 27 (2007), 1473-1508.
|
| [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.
|
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [26] |
S. Smale,
On gradient dynamical systems, Ann. Math.(2), 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. 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. |
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. |
| [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. Bonatti, S. Gan and L. Wen,
On the existence of non-trivial homoclinic classes, Ergodic Theory Dynam. Systems, 27 (2007), 1473-1508.
|
| [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.
|
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [26] |
S. Smale,
On gradient dynamical systems, Ann. Math.(2), 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. 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. |
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