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| A Characterization of Topologically Transitive Attributes for a Class of Dynamical Systems |
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Citation： |
Jiandong YIN,Zuoling ZHOU.A Characterization of Topologically Transitive Attributes for a Class of Dynamical Systems[J].Chinese Annals of Mathematics B,2012,33(3):419~428 |
| Page view： 1980
Net amount： 1273 |
Authors： |
Jiandong YIN; Zuoling ZHOU; |
Foundation： |
the National Natural Science Foundation of China (No. 10971236), the Foundation of Jiangxi Provincial Education Department (No.GJJ11295) and the Jiangxi Provincial Natural Science Foundation of China (No. 20114BAB201006) |
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| Abstract： |
In this work, by virtue of the properties of weakly almost periodic points of a dynamical system $(X, T)$ with at least two points, the authors prove that, if the measure center $M(T)$ of $T$ is the whole space, that is, $M(T)=X$, then the following statements are equivalent:
(1) $(X, T)$ is ergodic mixing;\hspace{2.5mm}
(2) $(X, T)$ is topologically double ergodic
(3) $(X, T)$ is weak mixing;\quad\hspace{2.6mm}
(4) $(X, T)$ is extremely scattering
(5) $(X, T)$ is strong scattering
(6) $(X\times X, T\times T)$ is strong scattering;
(7) $(X\times X, T\times T)$ is extremely scattering;
(8) For any subset $S$ of $\mathbb{N}$ with upper density 1, there is a $c$-dense $F_\sigma$-chaotic set with respect to $S$.
As an application,the authors show that, for the sub-shift $\sigma_A$ of finite type determined by a $k\times k$-$(0,1)$ matrix $A$, $\sigma_A$ is strong mixing if and only if $\sigma_A$ is totally transitive. |
Keywords： |
Weakly almost periodic point, Measure center, Topologically transitive attribute, Chaotic set |
Classification： |
58F10, 58F12 |
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