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Flows with the weak two-sided limit shadowing property
1. | Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Korea |
2. | Department of Mathematics, Chungnam National University, Daejeon 34134, Korea |
In this paper we study the weak two-sided limit shadowing for flows on a compact metric space which is different with the usual shadowing, two-sided limit shadowing and L-shadowing, and characterize the weak two-sided limit shadowing flows from the pointwise and measurable viewpoints. Moreover, we prove that if a flow $ \phi $ has the weak two-sided limit shadowing property on its chain recurrent $ CR(\phi) $ then the set $ CR(\phi) $ is decomposed by a finite number of closed invariant sets on which $ \phi $ is topologically transitive and has the two-sided limit shadowing property.
References:
[1] |
N. Aoki,
On the homeomorphisms with pseudo-orbit tracing property, Tokyo J. Math., 6 (1983), 329-334.
doi: 10.3836/tjm/1270213874. |
[2] |
J. Aponte, B. Carvalho and W. Cordeiro,
Suspensions of homeomorphisms with the two-sided limit shadowing property, Dyn. Syst., 35 (2020), 315-335.
doi: 10.1080/14689367.2019.1693506. |
[3] |
J. Aponte and H. Villavicencio,
Shadowable points for flows, J. Dyn. Control Syst., 24 (2018), 701-719.
doi: 10.1007/s10883-017-9381-8. |
[4] |
A. Artigue, B. Carvalho, W. Cordeiro and J. Vieitez,
Beyond topological hyperbolicity: The L-shadowing property, J. Differential Equations, 268 (2020), 3057-3080.
doi: 10.1016/j.jde.2019.09.052. |
[5] |
R. Bowen and P. Walters,
Expansive one-parameter flows, J. Differential Equations, 12 (1972), 180-193.
doi: 10.1016/0022-0396(72)90013-7. |
[6] |
B. Carvalho,
Hyperbolicity, transitivity and the two-sided limit shadowing property, Proc. Amer. Math Soc., 143 (2015), 657-666.
doi: 10.1090/S0002-9939-2014-12250-7. |
[7] |
B. Carvalho,
Product Anosov diffeomorphisms and the two-sided limit shadowing property, Proc. Amer. Math Soc., 146 (2018), 1151-1164.
doi: 10.1090/proc/13790. |
[8] |
B. Carvalho and W. Cordeiro,
$N$-expansive homeomorphisms with the shadowing property, J. Differential Equations, 261 (2016), 3734-3755.
doi: 10.1016/j.jde.2016.06.003. |
[9] |
B. Carvalho and D. Kwietniak,
On homeomorphisms with the two-sided limit shadowing property, J. Math Anal. Appl., 420 (2014), 801-813.
doi: 10.1016/j.jmaa.2014.06.011. |
[10] |
T. Das, K. Lee, D. Richeson and J. Wiseman,
Spectral decomposition for topologically Anosov homeomorphisms on noncompact and non-metrizable spaces, Topology Appl., 160 (2013), 149-158.
doi: 10.1016/j.topol.2012.10.010. |
[11] |
M. Dong, K. Lee and N. Nguyen,
Expanding measures for homeomorphisms with eventually shadowing property, J. Korean Math. Soc., 57 (2020), 935-955.
doi: 10.4134/JKMS.j190453. |
[12] |
W. Jung, N. Nguyen and Y. Yang,
Spectral decomposition for rescaling expansive flows with rescaled shadowing, Discrete Conti. Dyn. Syst., 40 (2020), 2267-2283.
doi: 10.3934/dcds.2020113. |
[13] |
N. Kawaguchi,
On the shadowing and limit shadowing properties, Fund. Math., 249 (2020), 21-35.
doi: 10.4064/fm552-4-2019. |
[14] |
M. Komuro,
One-parameter flows with the pseudo orbit tracing property, Monatsh. Math., 98 (1984), 219-253.
doi: 10.1007/BF01507750. |
[15] |
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. |
[16] |
H. Le, K. Lee and N. Nguyen,
Spectral decomposition and stability of mild expansive systems, Topol. Methods Nonlinear Anal., 56 (2020), 63-81.
|
[17] |
K. Lee,
Hyperbolic sets with the strong limit shadowing property, J. Inequal. Appl., 6 (2001), 507-517.
doi: 10.1155/S1025583401000315. |
[18] |
K. Lee and N. Nguyen,
Spectral decomposition and $\Omega$-stability of flows with expanding measures, J. Differential Equations, 269 (2020), 7574-7604.
doi: 10.1016/j.jde.2020.06.002. |
[19] |
K. Lee, N. Nguyen and Y. Yang,
Topological stability and spectral decomposition for homeomorphisms on noncompact spaces, Discrete Conti. Dyn. Syst., 38 (2018), 2487-2503.
doi: 10.3934/dcds.2018103. |
[20] |
P. Oprocha,
Transitivity, two-sided limit shadowing property and dense $\omega$-chaos, J. Korean Math. Soc., 51 (2014), 837-851.
doi: 10.4134/JKMS.2014.51.4.837. |
[21] |
S. Y. Pilyugin,
Sets of dynamical systems with various limit shadowing properties, J. Dynam. Differential Equations, 19 (2007), 747-775.
doi: 10.1007/s10884-007-9073-2. |
[22] |
S. Smale,
Differentiable dynamical systems, Bull. Amer. Math. Soc., 37 (1967), 747-817.
doi: 10.1090/S0002-9904-1967-11798-1. |
[23] |
R. F. Thomas,
Stability properties of one-parameter flows, Proc. London Math. Soc., 45 (1982), 479-505.
doi: 10.1112/plms/s3-45.3.479. |
[24] |
Y. Zhu, J. Zhang and Y. Guo,
Invariant properties of limit shadowing property, Appl. Math. J. Chinese Univ. Ser. B, 19 (2004), 279-287.
doi: 10.1007/s11766-004-0036-7. |
show all references
References:
[1] |
N. Aoki,
On the homeomorphisms with pseudo-orbit tracing property, Tokyo J. Math., 6 (1983), 329-334.
doi: 10.3836/tjm/1270213874. |
[2] |
J. Aponte, B. Carvalho and W. Cordeiro,
Suspensions of homeomorphisms with the two-sided limit shadowing property, Dyn. Syst., 35 (2020), 315-335.
doi: 10.1080/14689367.2019.1693506. |
[3] |
J. Aponte and H. Villavicencio,
Shadowable points for flows, J. Dyn. Control Syst., 24 (2018), 701-719.
doi: 10.1007/s10883-017-9381-8. |
[4] |
A. Artigue, B. Carvalho, W. Cordeiro and J. Vieitez,
Beyond topological hyperbolicity: The L-shadowing property, J. Differential Equations, 268 (2020), 3057-3080.
doi: 10.1016/j.jde.2019.09.052. |
[5] |
R. Bowen and P. Walters,
Expansive one-parameter flows, J. Differential Equations, 12 (1972), 180-193.
doi: 10.1016/0022-0396(72)90013-7. |
[6] |
B. Carvalho,
Hyperbolicity, transitivity and the two-sided limit shadowing property, Proc. Amer. Math Soc., 143 (2015), 657-666.
doi: 10.1090/S0002-9939-2014-12250-7. |
[7] |
B. Carvalho,
Product Anosov diffeomorphisms and the two-sided limit shadowing property, Proc. Amer. Math Soc., 146 (2018), 1151-1164.
doi: 10.1090/proc/13790. |
[8] |
B. Carvalho and W. Cordeiro,
$N$-expansive homeomorphisms with the shadowing property, J. Differential Equations, 261 (2016), 3734-3755.
doi: 10.1016/j.jde.2016.06.003. |
[9] |
B. Carvalho and D. Kwietniak,
On homeomorphisms with the two-sided limit shadowing property, J. Math Anal. Appl., 420 (2014), 801-813.
doi: 10.1016/j.jmaa.2014.06.011. |
[10] |
T. Das, K. Lee, D. Richeson and J. Wiseman,
Spectral decomposition for topologically Anosov homeomorphisms on noncompact and non-metrizable spaces, Topology Appl., 160 (2013), 149-158.
doi: 10.1016/j.topol.2012.10.010. |
[11] |
M. Dong, K. Lee and N. Nguyen,
Expanding measures for homeomorphisms with eventually shadowing property, J. Korean Math. Soc., 57 (2020), 935-955.
doi: 10.4134/JKMS.j190453. |
[12] |
W. Jung, N. Nguyen and Y. Yang,
Spectral decomposition for rescaling expansive flows with rescaled shadowing, Discrete Conti. Dyn. Syst., 40 (2020), 2267-2283.
doi: 10.3934/dcds.2020113. |
[13] |
N. Kawaguchi,
On the shadowing and limit shadowing properties, Fund. Math., 249 (2020), 21-35.
doi: 10.4064/fm552-4-2019. |
[14] |
M. Komuro,
One-parameter flows with the pseudo orbit tracing property, Monatsh. Math., 98 (1984), 219-253.
doi: 10.1007/BF01507750. |
[15] |
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. |
[16] |
H. Le, K. Lee and N. Nguyen,
Spectral decomposition and stability of mild expansive systems, Topol. Methods Nonlinear Anal., 56 (2020), 63-81.
|
[17] |
K. Lee,
Hyperbolic sets with the strong limit shadowing property, J. Inequal. Appl., 6 (2001), 507-517.
doi: 10.1155/S1025583401000315. |
[18] |
K. Lee and N. Nguyen,
Spectral decomposition and $\Omega$-stability of flows with expanding measures, J. Differential Equations, 269 (2020), 7574-7604.
doi: 10.1016/j.jde.2020.06.002. |
[19] |
K. Lee, N. Nguyen and Y. Yang,
Topological stability and spectral decomposition for homeomorphisms on noncompact spaces, Discrete Conti. Dyn. Syst., 38 (2018), 2487-2503.
doi: 10.3934/dcds.2018103. |
[20] |
P. Oprocha,
Transitivity, two-sided limit shadowing property and dense $\omega$-chaos, J. Korean Math. Soc., 51 (2014), 837-851.
doi: 10.4134/JKMS.2014.51.4.837. |
[21] |
S. Y. Pilyugin,
Sets of dynamical systems with various limit shadowing properties, J. Dynam. Differential Equations, 19 (2007), 747-775.
doi: 10.1007/s10884-007-9073-2. |
[22] |
S. Smale,
Differentiable dynamical systems, Bull. Amer. Math. Soc., 37 (1967), 747-817.
doi: 10.1090/S0002-9904-1967-11798-1. |
[23] |
R. F. Thomas,
Stability properties of one-parameter flows, Proc. London Math. Soc., 45 (1982), 479-505.
doi: 10.1112/plms/s3-45.3.479. |
[24] |
Y. Zhu, J. Zhang and Y. Guo,
Invariant properties of limit shadowing property, Appl. Math. J. Chinese Univ. Ser. B, 19 (2004), 279-287.
doi: 10.1007/s11766-004-0036-7. |
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