Expansivity and Cone-fields in Metric Spaces

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• 作者：?ukasz Struski (1)
  Jacek Tabor (1)
  
• 关键词：Cone ; field ; Hyperbolicity ; Expansive map ; Lyapunov function ; 37D20
• 刊名：Journal of Dynamics and Differential Equations
• 出版年：2014
• 出版时间：September 2014
• 年：2014
• 卷：26
• 期：3
• 页码：517-527
• 全文大小：158 KB
• 参考文献：1. Capiński, M.J., Zgliczyński, P.: Cone conditions and covering relations for normally hyperbolic invariant manifolds. Discrete and Continuous Dynamical Systems A., 641-70, (2011)
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• 作者单位：?ukasz Struski (1)
  Jacek Tabor (1)
  
  1. Faculty of Mathematics and Computer Science, ?ojasiewicza 6, 30-348?, Kraków, Poland
  
• ISSN：1572-9222

文摘

Due to the results of Lewowicz and Tolosa expansivity can be characterized with the aid of Lyapunov function. In this paper we study a similar problem for uniform expansivity and show that it can be described using generalized cone-fields on metric spaces. We say that a function \(f:X\rightarrow X\) is uniformly expansive on a set \(\varLambda \subset X\) if there exist \(\varepsilon >0\) and \(\alpha \in (0,1)\) such that for any two orbits \(\hbox {x}:\{-N,\ldots ,N\} \rightarrow \varLambda \) , \(\hbox {v}:\{-N,\ldots ,N\} \rightarrow X\) of \(f\) we have $$\begin{aligned} \sup _{-N\le n\le N}d(\hbox {x}_n,\hbox {v}_n) \le \varepsilon \implies d(\hbox {x}_0,\hbox {v}_0) \le \alpha \sup _{-N\le n\le N}d(\hbox {x}_n,\hbox {v}_n). \end{aligned}$$ It occurs that a function is uniformly expansive iff there exists a generalized cone-field on \(X\) such that \(f\) is cone-hyperbolic.