NDLI: Expansivity and Cone-fields in Metric Spaces

Content Provider	Springer Nature Link
Author	Tabor, Jacek Struski, Łukasz
Copyright Year	2014
Abstract	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.
Ending Page	527
Page Count	11
Starting Page	517
File Format	PDF
ISSN	10407294
e-ISSN	15729222
Journal	Journal of Dynamics and Differential Equations
Issue Number	3
Volume Number	26
Language	English
Publisher	Springer US
Publisher Date	2014-07-10
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) Ordinary Differential Equations Cone-field Hyperbolicity Expansive map Applications of Mathematics Partial Differential Equations Lyapunov function
Content Type	Text
Resource Type	Article
Subject	Analysis

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