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DCDS

Entropy sets are defined both topologically and for a
measure. The set of topological entropy sets is
the union of the sets of entropy sets for all invariant measures.
For a topological system $(X,T)$ and an invariant measure $\mu$ on
$(X,T)$, let $H(X,T)$ (resp. $H^\mu(X,T)$) be the closure of the
set of all entropy sets (resp. $\mu$- entropy sets) in the
hyperspace $2^X$. It is shown that if $h_{\text{top}}(T)>0$ (resp.
$h_\mu(T)>0$), the subsystem $(H(X,T),\hat{T})$ (resp.
$(H^\mu(X,T),\hat{T}))$ of $(2^X,\hat{T})$ has an
invariant measure with full support and infinite topological
entropy.

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 $H(X,T)$.

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 $H(X,T)$.

DCDS

The topological pressure is defined for
sub-additive potentials via separated sets and open covers
in general compact dynamical systems. A variational principle for
the topological pressure is set up without any additional
assumptions. The relations between different approaches in defining
the topological pressure are discussed. The result will have some potential applications in the multifractal analysis of iterated function systems with overlaps, the distribution of Lyapunov exponents and the dimension theory in dynamical systems.

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