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On global smooth solutions of 3-D compressible Euler equations with vanishing density in infinitely expanding balls
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 |
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.
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Weak attractor in flows on noncompact spaces, Dyn. Syst. Appl., 5 (1996), 503-519.
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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. |
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A rescaled expansiveness of flows, Trans. Amer. Math. Soc., 371 (2019), 3179-3207.
doi: 10.1090/tran/7382. |
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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. 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. |
| [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. Cordeiro, M. Denker and X. Zhang,
On specification and measure expansiveness, Discrete Contin. Dyn. Syst., 37 (2017), 1941-1957.
doi: 10.3934/dcds.2017082. |
| [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. |
| [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. 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. |
| [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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