
Abstract
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 $H(X,T)$.
Mathematics Subject Classification: Primary: 37B05; Secondary: 54H20.
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