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April  2020, 40(4): 2267-2283. doi: 10.3934/dcds.2020113

Spectral decomposition for rescaling expansive flows with rescaled shadowing

Woochul Jung 1, , Ngocthach Nguyen 1, and  Yinong Yang 2,,

1. 

Department of Mathematics, Chungnam National University, Daejeon 34134, Korea
2. 

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

* Corresponding author

Received  May 2019 Published  January 2020

In this paper, we introduce the concepts of rescaled expansiveness and the rescaled shadowing property for flows on metric spaces which are dynamical properties, and present a spectral decomposition theorem for flows. More precisely, we prove that if a flow is rescaling expansive and has the rescaled shadowing property on a locally compact metric space, then it admits the spectral decomposition. Moreover, we show that if a flow on locally compact metric space has the rescaled shadowing property then its restriction on nonwandering set also has the rescaled shadowing property.

Keywords: Rescaled expansiveness, rescaled shadowing property, spectral decomposition.
Mathematics Subject Classification: Primary: 37Bxx; Secondary: 37Dxx.
Citation: Woochul Jung, Ngocthach Nguyen, Yinong Yang. Spectral decomposition for rescaling expansive flows with rescaled shadowing. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2267-2283. doi: 10.3934/dcds.2020113
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]

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

[3]

A. Artigue, Rescaled expansivity and separating flows, Discrete Contin. Dyn. Syst., 38 (2018), 4433-4447.  doi: 10.3934/dcds.2018193.  Google Scholar

[4]

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

[5]

W. Cordeiro, M. Denker and X. Zhang, On specification and measure expansiveness, Discrete Contin. Dyn. Syst., 37 (2017), 1941-1957.  doi: 10.3934/dcds.2017082.  Google Scholar

[6]

T. Das, K. Lee, D. Richeson and J. Wiseman, Spectral decomposition for topologically Anosov homemorphisms on noncompact and non-metrizable spaces, Topology Appl., 160 (2013), 149-158.  doi: 10.1016/j.topol.2012.10.010.  Google Scholar

[7]

M. Hurley, Chain recurrence, semiflows, and gradients, J. Dynam. Differential Equations, 7 (1995), 437-456.  doi: 10.1007/BF02219371.  Google Scholar

[8]

M. Komuro, Expansive properties of Lorenz attractors, The Theory of Dynamical Systems and Its Applications to Nonlinear Problems, WWorld Sci. Publishing, Singapore, (1984), 4–26.  Google Scholar

[9]

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

[10]

K.-H. Lee, Weak attractor in flows on noncompact spaces, Dyn. Syst. Appl., 5 (1996), 503-519.   Google Scholar

[11]

K. Lee, N.-T. Nguyen and Y. N. Yang, Topological stability and spectral decomposition for homeomophisms on noncompact spaces, Discrete Contin. Dyn. Syst., 38 (2018), 2487-2503.  doi: 10.3934/dcds.2018103.  Google Scholar

[12]

Z. Nitecki, Explosions in completely unstable flows. Ⅰ. Preventing explosions, Trans. Amer. Math. Soc., 245 (1978), 43-61.  doi: 10.2307/1998856.  Google Scholar

[13]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

[14]

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

[15]

X. Wen and Y. N. Yu, Equivalent definitions of rescaled expansiveness, J. Korean Math. Soc., 55 (2018), 593-604.   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]

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

[3]

A. Artigue, Rescaled expansivity and separating flows, Discrete Contin. Dyn. Syst., 38 (2018), 4433-4447.  doi: 10.3934/dcds.2018193.  Google Scholar

[4]

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

[5]

W. Cordeiro, M. Denker and X. Zhang, On specification and measure expansiveness, Discrete Contin. Dyn. Syst., 37 (2017), 1941-1957.  doi: 10.3934/dcds.2017082.  Google Scholar

[6]

T. Das, K. Lee, D. Richeson and J. Wiseman, Spectral decomposition for topologically Anosov homemorphisms on noncompact and non-metrizable spaces, Topology Appl., 160 (2013), 149-158.  doi: 10.1016/j.topol.2012.10.010.  Google Scholar

[7]

M. Hurley, Chain recurrence, semiflows, and gradients, J. Dynam. Differential Equations, 7 (1995), 437-456.  doi: 10.1007/BF02219371.  Google Scholar

[8]

M. Komuro, Expansive properties of Lorenz attractors, The Theory of Dynamical Systems and Its Applications to Nonlinear Problems, WWorld Sci. Publishing, Singapore, (1984), 4–26.  Google Scholar

[9]

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

[10]

K.-H. Lee, Weak attractor in flows on noncompact spaces, Dyn. Syst. Appl., 5 (1996), 503-519.   Google Scholar

[11]

K. Lee, N.-T. Nguyen and Y. N. Yang, Topological stability and spectral decomposition for homeomophisms on noncompact spaces, Discrete Contin. Dyn. Syst., 38 (2018), 2487-2503.  doi: 10.3934/dcds.2018103.  Google Scholar

[12]

Z. Nitecki, Explosions in completely unstable flows. Ⅰ. Preventing explosions, Trans. Amer. Math. Soc., 245 (1978), 43-61.  doi: 10.2307/1998856.  Google Scholar

[13]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

[14]

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

[15]

X. Wen and Y. N. Yu, Equivalent definitions of rescaled expansiveness, J. Korean Math. Soc., 55 (2018), 593-604.   Google Scholar

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