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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 ParisEst), 5 boulevard Descartes, 77454 MarnelaVallée 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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