The Research of G- expansive Map and Devaney’s G-Chaos Condition on Metric G-Space

Abstract

The expansive map and chaos have an important significance in terms of theory and application. According to the definition of expansive map and chaos, we give the concept of G-expansive map and Devaney's G-Chaos in this paper. By inference, some conclusions of the metric space were extended to the metric G-space. We have the following result:(1) Let X be a metric G-space, G be a commutative topological group and f be a equivariant self-map on X .Then we have that the map f satisfies the first condition and second condition of Devaney's G-Chaos if and only if the sets U and V have the same periodic orbit where U and V be any nonempty open sets; (2) Let X be a metric G-space and f be a homeomorphism map of X into itself. Then we have that f is a G-expansive map if and only if f^m is a Gexpansive map for any nonzero integer m; (3) Let (X,d_1) and (Y,d_2 ) be a metric G-space. Let f be a homeomorphism and equivariant map from X to Y. If h is a G-expansive map of X into itself, then fhf^-11 is a Gexpansive map of Y into itself; (4) Let X be a metric G-space, A be a invariant subset of G and X X-A be finite. Let f be a homeomorphism map on X. If f is G-expansive on A, then f is G-expansive on X; (5) Let X be a compact metric G-space and G be compact. If f is a G-expansive map with G-expansive constant δ, then for each ε > 0 satifying 0 < ε <δ there exists a positive integer k such that d(G(x),G(y)) ε implies d( f^n(G(x)), f^n(G(y))) > δ for some integer n where |n|

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度量G空间上G-膨胀映射和Devaney G-混沌条件的研究

膨胀图和混沌具有重要的理论和应用意义。根据扩张映射和混沌的定义，给出了g -扩张映射和Devaney的g -混沌的概念。通过推理，将度量空间的一些结论推广到度量g空间。(1)设X是度量G空间，G是交换拓扑群，f是X上的等变自映射，则映射f满足Devaney G- chaos的第一条件和第二条件，当且仅当集合U和V具有相同的周期轨道，其中U和V为任意非空开集;(2)设X是度量g空间，f是X到自身的同胚映射。那么我们有f是g扩张映射当且仅当f^m是任意非零整数m的g扩张映射;(3)设(X,d_1)和(Y,d_2)是一个度量g空间。设f是X到Y的同胚等变映射，如果h是X到自身的g扩张映射，则fhf^-11是Y到自身的g扩张映射;(4)设X是度量G空间，a是G的不变子集，X X- a是有限的。设f是X上的同胚映射，如果f在a上是g扩张的，那么f在X上是g扩张的;(5)设X是紧度量G空间，G是紧的。如果f是一个具有g膨胀常数δ的g膨胀映射，则对于某整数n| n| < ‰·k，对于每个ε> 0满足0 < εδ，这些结果丰富了g膨胀映射和Devaney的G-Chaos理论。

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