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Stability from the point of view of diffusion, relaxation and spatial inhomogeneity
Entropy sets, weakly mixing sets and entropy capacity
1. | LAMA (CNRS and Université Paris-Est), 5 boulevard Descartes, 77454 Marne-la-Vallée cedex 2, France |
2. | Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China |
Weakly mixing sets and partial mixing of dynamical systems are introduced and characterized. It is proved that if $h_{\text{top}}(T)>0$ (resp. $h_\mu(T)>0$) the set of all weakly mixing entropy sets (resp. $\mu$-entropy sets) is a dense $G_\delta$ in $H(X,T)$ (resp. $H^\mu(X,T)$). A Devaney chaotic but not partly mixing system is constructed.
Concerning entropy capacities, it is shown that when $\mu$ is ergodic with $h_\mu(T)>0$, the set of all weakly mixing $\mu$-entropy sets $E$ such that the Bowen entropy $h(E)\ge h_\mu(T)$ is residual in $H^\mu(X,T)$. When in addition $(X,T)$ is uniquely ergodic the set of all weakly mixing entropy sets $E$ with $h(E)=h_{\text{top}}(T)$ is residual in $H(X,T)$.
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