# American Institute of Mathematical Sciences

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

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