-
Previous Article
Integrability of moduli and regularity of denjoy counterexamples
- DCDS Home
- This Issue
-
Next Article
Nonuniformly hyperbolic systems arising from coupling of chaotic and gradient-like systems
No-shadowing for singular hyperbolic sets with a singularity
1. | School of Mathematical Sciences, Beihang University, Beijing 100191, China |
2. | School of Mathematical Sciences, Peking University, Beijing 100871, China |
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.
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. |
[2] |
C. Bonatti and A. da Luz, Star flows and multisingular hyperbolicity, preprint, arXiv: 1705.05799. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
S. T. Liao,
A basic property of a certain class of differential systems (in Chinese), Acta Math. Sinica, 22 (1979), 316-343.
|
[9] |
R. Mañé,
Contributions to the stablity conjecture, Topology, 17 (1978), 383-396.
doi: 10.1016/0040-9383(78)90005-8. |
[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. |
[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. |
[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. |
[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. |
[14] |
X. Wen and L. Wen,
A rescaled expansiveness for flows, Trans. Amer. Math. Soc., 371 (2019), 3179-3207.
doi: 10.1090/tran/7382. |
[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. |
[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. |
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. |
[2] |
C. Bonatti and A. da Luz, Star flows and multisingular hyperbolicity, preprint, arXiv: 1705.05799. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
S. T. Liao,
A basic property of a certain class of differential systems (in Chinese), Acta Math. Sinica, 22 (1979), 316-343.
|
[9] |
R. Mañé,
Contributions to the stablity conjecture, Topology, 17 (1978), 383-396.
doi: 10.1016/0040-9383(78)90005-8. |
[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. |
[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. |
[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. |
[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. |
[14] |
X. Wen and L. Wen,
A rescaled expansiveness for flows, Trans. Amer. Math. Soc., 371 (2019), 3179-3207.
doi: 10.1090/tran/7382. |
[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. |
[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. |
[1] |
Yi Shi, Shaobo Gan, Lan Wen. On the singular-hyperbolicity of star flows. Journal of Modern Dynamics, 2014, 8 (2) : 191-219. doi: 10.3934/jmd.2014.8.191 |
[2] |
Sergey Kryzhevich, Sergey Tikhomirov. Partial hyperbolicity and central shadowing. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2901-2909. doi: 10.3934/dcds.2013.33.2901 |
[3] |
Marcin Mazur, Jacek Tabor, Piotr Kościelniak. Semi-hyperbolicity and hyperbolicity. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 1029-1038. doi: 10.3934/dcds.2008.20.1029 |
[4] |
Raquel Ribeiro. Hyperbolicity and types of shadowing for $C^1$ generic vector fields. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2963-2982. doi: 10.3934/dcds.2014.34.2963 |
[5] |
Marcin Mazur, Jacek Tabor. Computational hyperbolicity. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1175-1189. doi: 10.3934/dcds.2011.29.1175 |
[6] |
Boris Hasselblatt, Yakov Pesin, Jörg Schmeling. Pointwise hyperbolicity implies uniform hyperbolicity. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2819-2827. doi: 10.3934/dcds.2014.34.2819 |
[7] |
Jihoon Lee, Ngocthach Nguyen. Flows with the weak two-sided limit shadowing property. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4375-4395. doi: 10.3934/dcds.2021040 |
[8] |
Luis Barreira, Claudia Valls. Growth rates and nonuniform hyperbolicity. Discrete and Continuous Dynamical Systems, 2008, 22 (3) : 509-528. doi: 10.3934/dcds.2008.22.509 |
[9] |
Rasul Shafikov, Christian Wolf. Stable sets, hyperbolicity and dimension. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 403-412. doi: 10.3934/dcds.2005.12.403 |
[10] |
Arno Berger. On finite-time hyperbolicity. Communications on Pure and Applied Analysis, 2011, 10 (3) : 963-981. doi: 10.3934/cpaa.2011.10.963 |
[11] |
Christian Bonatti, Shaobo Gan, Dawei Yang. On the hyperbolicity of homoclinic classes. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1143-1162. doi: 10.3934/dcds.2009.25.1143 |
[12] |
Mickaël Kourganoff. Uniform hyperbolicity in nonflat billiards. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1145-1160. doi: 10.3934/dcds.2018048 |
[13] |
Andy Hammerlindl, Jana Rodriguez Hertz, Raúl Ures. Ergodicity and partial hyperbolicity on Seifert manifolds. Journal of Modern Dynamics, 2020, 0: 331-348. doi: 10.3934/jmd.2020012 |
[14] |
Zhenning Cai, Yuwei Fan, Ruo Li. On hyperbolicity of 13-moment system. Kinetic and Related Models, 2014, 7 (3) : 415-432. doi: 10.3934/krm.2014.7.415 |
[15] |
Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz, Raúl Ures. Partial hyperbolicity and ergodicity in dimension three. Journal of Modern Dynamics, 2008, 2 (2) : 187-208. doi: 10.3934/jmd.2008.2.187 |
[16] |
Jana Rodriguez Hertz. Genericity of nonuniform hyperbolicity in dimension 3. Journal of Modern Dynamics, 2012, 6 (1) : 121-138. doi: 10.3934/jmd.2012.6.121 |
[17] |
Jérôme Buzzi, Todd Fisher. Entropic stability beyond partial hyperbolicity. Journal of Modern Dynamics, 2013, 7 (4) : 527-552. doi: 10.3934/jmd.2013.7.527 |
[18] |
Yakov Pesin. On the work of Dolgopyat on partial and nonuniform hyperbolicity. Journal of Modern Dynamics, 2010, 4 (2) : 227-241. doi: 10.3934/jmd.2010.4.227 |
[19] |
Manseob Lee, Jumi Oh, Xiao Wen. Diffeomorphisms with a generalized Lipschitz shadowing property. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1913-1927. doi: 10.3934/dcds.2020346 |
[20] |
Fang Zhang, Yunhua Zhou. On the limit quasi-shadowing property. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2861-2879. doi: 10.3934/dcds.2017123 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]