2,354 research outputs found
Lowering topological entropy over subsets revisited
Let be a topological dynamical system. Denote by and the covering entropy and dimensional entropy of ,
respectively. is called D-{\it lowerable} (resp. {\it lowerable}) if
for each there is a subset (resp. closed subset)
with (resp. ); is called D-{\it hereditarily
lowerable} (resp. {\it hereditarily lowerable}) if each Souslin subset (resp.
closed subset) is D-lowerable (resp. lowerable).
In this paper it is proved that each topological dynamical system is not only
lowerable but also D-lowerable, and each asymptotically -expansive system is
D-hereditarily lowerable. A minimal system which is lowerable and not
hereditarily lowerable is demonstrated.Comment: All comments are welcome. Transactions of the American Mathematical
Society, to appea
Local entropy theory for a countable discrete amenable group action
In the paper we throw the first light on studying systematically the local
entropy theory for a countable discrete amenable group action. For such an
action, we introduce entropy tuples in both topological and measure-theoretic
settings and build the variational relation between these two kinds of entropy
tuples by establishing a local variational principle for a given finite open
cover. Moreover, based the idea of topological entropy pairs, we introduce and
study two special classes of such an action: uniformly positive entropy and
completely positive entropy. Note that in the building of the local variational
principle, following Romagnoli's ideas two kinds of measure-theoretic entropy
are introduced for finite Borel covers. These two kinds of entropy turn out to
be the same, where Danilenko's orbital approach becomes an inevitable tool
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