45,936 research outputs found
Existence of stable solutions to in with and
We consider the polyharmonic equation in
with and . We prove the existence of many entire stable
solutions. This answer some questions raised by Farina and Ferrero
Pointwise convergence of multiple ergodic averages and strictly ergodic models
By building some suitable strictly ergodic models, we prove that for an
ergodic system , , , the averages converge a.e.
Deriving some results from the construction, for distal systems we answer
positively the question if the multiple ergodic averages converge a.e. That is,
we show that if is an ergodic distal system, and , then multiple ergodic averages converge a.e.Comment: 35 pages, revised version following referees' report
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
Stable sets and mean Li-Yorke chaos in positive entropy systems
It is shown that in a topological dynamical system with positive entropy,
there is a measure-theoretically "rather big" set such that a multivariant
version of mean Li-Yorke chaos happens on the closure of the stable or unstable
set of any point from the set. It is also proved that the intersections of the
sets of asymptotic tuples and mean Li-Yorke tuples with the set of topological
entropy tuples are dense in the set of topological entropy tuples respectively.Comment: The final version, reference updated, to appear in Journal of
Functional Analysi
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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