28 research outputs found
A Birkhoff type transitivity theorem for non-separable completely metrizable spaces with applications to Linear Dynamics
In this note we prove a Birkhoff type transitivity theorem for continuous
maps acting on non-separable completely metrizable spaces and we give some
applications for dynamics of bounded linear operators acting on complex
Fr\'{e}chet spaces. Among them we show that any positive power and any
unimodular multiple of a topologically transitive linear operator is
topologically transitive, generalizing similar results of S.I. Ansari and F.
Le\'{o}n-Saavedra V. M\"{u}ller for hypercyclic operators.Comment: Several changes concerning the presentation of the paper; title
changed; 12 page
The group of isometries of a locally compact metric space with one end
In this note we study the dynamics of the natural evaluation action of the
group of isometries of a locally compact metric space with one end.
Using the notion of pseudo-components introduced by S. Gao and A. S. Kechris we
show that has only finitely many pseudo-components of which exactly one is
not compact and acts properly on. The complement of the non-compact
component is a compact subset of and may fail to act properly on it.Comment: Submitted for publication, 7 page
Limit and extended limit sets of matrices in Jordan normal form
In this note we describe the limit and the extended limit sets of every
vector for a single matrix in Jordan normal form.Comment: 10 pages, we corrected some typos and we added a questio
J-class weighted shifts on the space of bounded sequences of complex numbers
We provide a characterization of -class and -class unilateral
weighted shifts on in terms of their weight sequences.
In contrast to the previously mentioned result we show that a bilateral
weighted shift on cannot be a -class operator.Comment: We correct some of the statements and the proof
On embeddings of proper and equicontinuous actions in zero-dimensional compactifications
We provide a tool for studying properly discontinuous actions of non-compact
groups on locally compact, connected and paracompact spaces, by embedding such
an action in a suitable zero-dimensional compactification of the underlying
space with pleasant properties. Precisely, given such an action we
construct a zero-dimensional compactification of with the
properties: (a) there exists an extension of the action on , (b) if is the set of the limit points of the orbits of
the initial action in , then the restricted action remains properly discontinuous, is indivisible and equicontinuous with
respect to the uniformity induced on by that of ,
and (c) is the maximal among the zero-dimensional compactifications of
with these properties. Proper actions are usually embedded in the end point
compactification of , in order to obtain topological invariants
concerning the cardinality of the space of the ends of , provided that
has an additional "nice" property of rather local character ("property Z",
i.e., every compact subset of is contained in a compact and connected one).
If the considered space has this property, our new compactification coincides
with the end point one. On the other hand, we give an example of a space not
having the "property Z" for which our compactification is different from the
end point compactification. As an application, we show that the invariant
concerning the cardinality of the ends of holds also for a class of actions
strictly containing the properly discontinuous ones and for spaces not
necessarily having "property Z".Comment: 18 page
Proper actions and proper invariant metrics
We show that if a (locally compact) group acts properly on a locally
compact -compact space then there is a family of -invariant
proper continuous finite-valued pseudometrics which induces the topology of
. If is furthermore metrizable then acts properly on if and only
if there exists a -invariant proper compatible metric on .Comment: The paper has been completely rewritten and differs essentially from
"Constructing invariant Heine-Borel metrics for proper G-spaces". The main
result extended to the more general case when is a topological group
which acts properly on a locally compact -compact Hausdorff space
. Note that there is a gap in the proof of Theorem 2.4 of the old versio
Dynamics of tuples of matrices
In this article we answer a question raised by N. Feldman in \cite{Feldman}
concerning the dynamics of tuples of operators on . In
particular, we prove that for every positive integer there exist
tuples of matrices over such
that is hypercyclic. We also establish related results
for tuples of matrices over or being in
Jordan form.Comment: 10 page
The Jacobson radical for analytic crossed products
We characterise the (Jacobson) radical of the analytic crossed product of
C_0(X) by the non-negative integers (Z_+), answering a question first raised by
Arveson and Josephson in 1969. In fact, we characterise the radical of analytic
crossed products of C_0(X) by (Z_+)^d. The radical consists of all elements
whose `Fourier coefficients' vanish on the recurrent points of the dynamical
system (and the first one is zero). The multi-dimensional version requires a
variation of the notion of recurrence, taking into account the various degrees
of freedom.Comment: 17 pages; AMS-LaTeX; minor correction
Topological generators of abelian Lie groups and hypercyclic finitely generated abelian semigroups of matrices
In this paper we bring together results about the density of subsemigroups of
abelian Lie groups, the minimal number of topological generators of abelian Lie
groups and a result about actions of algebraic groups. We find the minimal
number of generators of a finitely generated abelian semigroup or group of
matrices with a dense or a somewhere dense orbit by computing the minimal
number of generators of a dense subsemigroup (or subgroup) of the connected
component of the identity of its Zariski closure.Comment: 14 page