Oriented Shadowing Property and \(\Omega \)-Stability for Vector Fields
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We call that a vector field has the oriented shadowing property if for any \(\varepsilon >0\) there is \(d>0\) such that each \(d\)-pseudo orbit is \(\varepsilon \)-oriented shadowed by some real orbit. In this paper, we show that the \(C^1\)-interior of the set of vector fields with the oriented shadowing property is contained in the set of vector fields with the \(\Omega \)-stability.
KeywordsVector fields Oriented shadowing \(\Omega \)-stability
Mathematics Subject Classification37C50
Shaobo Gan is supported by 973 project 2011CB808002, NSFC 11025101 and 11231001. Ming Li is partially supported by the National Science Foundation of China (Grant No. 11201244), the Specialized Research Fund for the Doctoral Program of Higher Education of China (SRFDP, Grant No. 20120031120024), and the LPMC of Nankai University. Sergey Tikhomirov is partially supported by Chebyshev Laboratory (Department of Mathematics and Mechanics, St. Petersburg State University) under RF Government grant 11.G34.31.0026, JSC “Gazprom neft”, by the Saint-Petersburg State University research grant 220.127.116.114 and by the German-Russian Interdisciplinary Science Center (G-RISC) funded by the German Federal Foreign Office via the German Academic Exchange Service (DAAD).
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