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October  2020, 40(10): 6043-6059. doi: 10.3934/dcds.2020258

No-shadowing for singular hyperbolic sets with a singularity

Xiao Wen 1, and  Lan Wen 2,

1. 

School of Mathematical Sciences, Beihang University, Beijing 100191, China
2. 

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

Received  March 2020 Published  June 2020

Fund Project: The first author is supported by National Natural Science Foundation of China (No. 11671025 and No. 11571188) and the Fundamental Research Funds for the Central Universities. The second author is supported by National Natural Science Foundation of China (No. 11231001)

We prove that every singular hyperbolic chain transitive set with a singularity does not admit the shadowing property. Using this result we show that if a star flow has the shadowing property on its chain recurrent set then it satisfies Axiom A and the no-cycle conditions; and that if a multisingular hyperbolic set has the shadowing property then it is hyperbolic.

Keywords: Shadowing property, singular hyperbolicity, star flows, multisingular hyperbolicity.
Mathematics Subject Classification: Primary: 37C10, 37D30, 37C50.
Citation: Xiao Wen, Lan Wen. No-shadowing for singular hyperbolic sets with a singularity. Discrete & Continuous Dynamical Systems, 2020, 40 (10) : 6043-6059. doi: 10.3934/dcds.2020258
References:
[1]

V. Araujo, M. J. Pacifico, E. R. Pujals and M. Viana, Singular-hyperbolic attractors are chaotic, Trans. Amer. Math. Soc., 361 (2009), 2431-2485.  doi: 10.1090/S0002-9947-08-04595-9.  Google Scholar

[2]

C. Bonatti and A. da Luz, Star flows and multisingular hyperbolicity, preprint, arXiv: 1705.05799. Google Scholar

[3]

S. Crovisier, A. da Luz, D. Yang and J. Zhang, On the notions of singular domination and (multi-)singular Hyperbolicity, preprint, arXiv: 2003.07099. Google Scholar

[4]

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

[5]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977. doi: 10.1007/BFb0092042.  Google Scholar

[6]

M. Komuro, Lorenz attractors do not have the pseudo-orbit tracing property, J. Math. Soc. Japan, 37 (1985), 489-514.  doi: 10.2969/jmsj/03730489.  Google Scholar

[7]

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

[8]

S. T. Liao, A basic property of a certain class of differential systems (in Chinese), Acta Math. Sinica, 22 (1979), 316-343.   Google Scholar

[9]

R. Mañé, Contributions to the stablity conjecture, Topology, 17 (1978), 383-396.  doi: 10.1016/0040-9383(78)90005-8.  Google Scholar

[10]

C. A. Morales and M. J. Pacifico, A dichotomy for three-dimensional vector fields, Ergodic Theory Dynam. Systems, 23 (2003), 1575-1600.  doi: 10.1017/S0143385702001621.  Google Scholar

[11]

C. A. Morales, M. J. Pacifico and E. R. Pujals, Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers, Ann. of Math. (2), 160 (2004), 375–432. doi: 10.4007/annals.2004.160.375.  Google Scholar

[12]

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

[13]

N. Sumi, P. Varandas and K. Yamamoto, Specification and partial hyperbolicity for flows, Dyn. Syst., 30 (2015), 501-524.  doi: 10.1080/14689367.2015.1081380.  Google Scholar

[14]

X. Wen and L. Wen, A rescaled expansiveness for flows, Trans. Amer. Math. Soc., 371 (2019), 3179-3207.  doi: 10.1090/tran/7382.  Google Scholar

[15]

X. Wen, L. Wen and D. Yang, A characterization of singular hyperbolicity via the linear Poincaré flow, J. Differential Equations, 268 (2020), 4256-4275.  doi: 10.1016/j.jde.2019.10.029.  Google Scholar

[16]

T. Yamanaka, A characterization of dense vector fields in $\eth^1(M)$ on 3-manifolds, Hokkaido Math. J., 31 (2002), 97-105.  doi: 10.14492/hokmj/1350911772.  Google Scholar

show all references

References:
[1]

V. Araujo, M. J. Pacifico, E. R. Pujals and M. Viana, Singular-hyperbolic attractors are chaotic, Trans. Amer. Math. Soc., 361 (2009), 2431-2485.  doi: 10.1090/S0002-9947-08-04595-9.  Google Scholar

[2]

C. Bonatti and A. da Luz, Star flows and multisingular hyperbolicity, preprint, arXiv: 1705.05799. Google Scholar

[3]

S. Crovisier, A. da Luz, D. Yang and J. Zhang, On the notions of singular domination and (multi-)singular Hyperbolicity, preprint, arXiv: 2003.07099. Google Scholar

[4]

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

[5]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977. doi: 10.1007/BFb0092042.  Google Scholar

[6]

M. Komuro, Lorenz attractors do not have the pseudo-orbit tracing property, J. Math. Soc. Japan, 37 (1985), 489-514.  doi: 10.2969/jmsj/03730489.  Google Scholar

[7]

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

[8]

S. T. Liao, A basic property of a certain class of differential systems (in Chinese), Acta Math. Sinica, 22 (1979), 316-343.   Google Scholar

[9]

R. Mañé, Contributions to the stablity conjecture, Topology, 17 (1978), 383-396.  doi: 10.1016/0040-9383(78)90005-8.  Google Scholar

[10]

C. A. Morales and M. J. Pacifico, A dichotomy for three-dimensional vector fields, Ergodic Theory Dynam. Systems, 23 (2003), 1575-1600.  doi: 10.1017/S0143385702001621.  Google Scholar

[11]

C. A. Morales, M. J. Pacifico and E. R. Pujals, Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers, Ann. of Math. (2), 160 (2004), 375–432. doi: 10.4007/annals.2004.160.375.  Google Scholar

[12]

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

[13]

N. Sumi, P. Varandas and K. Yamamoto, Specification and partial hyperbolicity for flows, Dyn. Syst., 30 (2015), 501-524.  doi: 10.1080/14689367.2015.1081380.  Google Scholar

[14]

X. Wen and L. Wen, A rescaled expansiveness for flows, Trans. Amer. Math. Soc., 371 (2019), 3179-3207.  doi: 10.1090/tran/7382.  Google Scholar

[15]

X. Wen, L. Wen and D. Yang, A characterization of singular hyperbolicity via the linear Poincaré flow, J. Differential Equations, 268 (2020), 4256-4275.  doi: 10.1016/j.jde.2019.10.029.  Google Scholar

[16]

T. Yamanaka, A characterization of dense vector fields in $\eth^1(M)$ on 3-manifolds, Hokkaido Math. J., 31 (2002), 97-105.  doi: 10.14492/hokmj/1350911772.  Google Scholar

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