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

Spectral decomposition for rescaling expansive flows with rescaled shadowing

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.

Citation: Woochul Jung, Ngocthach Nguyen, Yinong Yang. Spectral decomposition for rescaling expansive flows with rescaled shadowing. Discrete and Continuous Dynamical Systems, 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.

[2]

V. AraujoM. J. PacificoE. 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.

[3]

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

[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.

[5]

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

[6]

T. DasK. LeeD. 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.

[7]

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

[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.

[9]

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

[10]

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

[11]

K. LeeN.-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.

[12]

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

[13]

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

[14]

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

[15]

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

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]

V. AraujoM. J. PacificoE. 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.

[3]

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

[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.

[5]

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

[6]

T. DasK. LeeD. 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.

[7]

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

[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.

[9]

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

[10]

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

[11]

K. LeeN.-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.

[12]

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

[13]

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

[14]

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

[15]

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

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