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September  2021, 41(9): 4375-4395. doi: 10.3934/dcds.2021040

Flows with the weak two-sided limit shadowing property

Jihoon Lee 1, and  Ngocthach Nguyen 2,,

1. 

Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Korea
2. 

Department of Mathematics, Chungnam National University, Daejeon 34134, Korea

* Corresponding author: Ngocthach Nguyen

Received  April 2020 Revised  August 2020 Published  September 2021 Early access  March 2021

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.

Keywords: Expansiveness, shadowing property, spectral decomposition, transitivity, weak two-sided limit shadowing property.
Mathematics Subject Classification: Primary: 37B99; Secondary: 37D99.
Citation: Jihoon Lee, Ngocthach Nguyen. Flows with the weak two-sided limit shadowing property. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4375-4395. doi: 10.3934/dcds.2021040
References:
[1]

N. Aoki, On the homeomorphisms with pseudo-orbit tracing property, Tokyo J. Math., 6 (1983), 329-334.  doi: 10.3836/tjm/1270213874.  Google Scholar

[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.  Google Scholar

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J. Aponte and H. Villavicencio, Shadowable points for flows, J. Dyn. Control Syst., 24 (2018), 701-719.  doi: 10.1007/s10883-017-9381-8.  Google Scholar

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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.  Google Scholar

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R. Bowen and P. Walters, Expansive one-parameter flows, J. Differential Equations, 12 (1972), 180-193.  doi: 10.1016/0022-0396(72)90013-7.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

N. Kawaguchi, On the shadowing and limit shadowing properties, Fund. Math., 249 (2020), 21-35.  doi: 10.4064/fm552-4-2019.  Google Scholar

[14]

M. Komuro, One-parameter flows with the pseudo orbit tracing property, Monatsh. Math., 98 (1984), 219-253.  doi: 10.1007/BF01507750.  Google Scholar

[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.  Google Scholar

[16]

H. Le, K. Lee and N. Nguyen, Spectral decomposition and stability of mild expansive systems, Topol. Methods Nonlinear Anal., 56 (2020), 63-81.   Google Scholar

[17]

K. Lee, Hyperbolic sets with the strong limit shadowing property, J. Inequal. Appl., 6 (2001), 507-517.  doi: 10.1155/S1025583401000315.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 37 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

N. Kawaguchi, On the shadowing and limit shadowing properties, Fund. Math., 249 (2020), 21-35.  doi: 10.4064/fm552-4-2019.  Google Scholar

[14]

M. Komuro, One-parameter flows with the pseudo orbit tracing property, Monatsh. Math., 98 (1984), 219-253.  doi: 10.1007/BF01507750.  Google Scholar

[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.  Google Scholar

[16]

H. Le, K. Lee and N. Nguyen, Spectral decomposition and stability of mild expansive systems, Topol. Methods Nonlinear Anal., 56 (2020), 63-81.   Google Scholar

[17]

K. Lee, Hyperbolic sets with the strong limit shadowing property, J. Inequal. Appl., 6 (2001), 507-517.  doi: 10.1155/S1025583401000315.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 37 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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