47 research outputs found
Some supports of Fourier transforms of singular measures are not Rajchman
The notion of Riesz sets tells us that a support of Fourier transform of a
measure with non-trivial singular part has to be large. The notion of Rajchman
sets tells us that if the Fourier transform tends to zero at infinity outside a
small set, then it tends to zero even on the small set. Here we present a new
angle of an old question: Whether every Rajchman set should be Riesz
An example of a minimal action of the free semi-group \F^{+}_{2} on the Hilbert space
The Invariant Subset Problem on the Hilbert space is to know whether there
exists a bounded linear operator on a separable infinite-dimensional
Hilbert space such that the orbit of every non-zero
vector under the action of is dense in . We show that there
exists a bounded linear operator on a complex separable
infinite-dimensional Hilbert space and a unitary operator on , such
that the following property holds true: for every non-zero vector ,
either or has a dense orbit under the action of . As a consequence,
we obtain in particular that there exists a minimal action of the free
semi-group with two generators \F^{+}_{2} on a complex separable
infinite-dimensional Hilbert space .Comment: 10
A general approach to Read's type constructions of operators without non-trivial invariant closed subspaces
We present a general method for constructing operators without non-trivial
invariant closed subsets on a large class of non-reflexive Banach spaces. In
particular, our approach unifies and generalizes several constructions due to
Read of operators without non-trivial invariant subspaces on the spaces
, or , and without non-trivial invariant
subsets on . We also investigate how far our methods can be extended
to the Hilbertian setting, and construct an operator on a quasireflexive dual
Banach space which has no non-trivial -closed invariant subspace.Comment: Minor modification
Conjugacy of real diffeomorphisms. A survey
Given a group G, the conjugacy problem in G is the problem of giving an
effective procedure for determining whether or not two given elements f, g of G
are conjugate, i.e. whether there exists h belonging to G with fh = hg. This
paper is about the conjugacy problem in the group Diffeo(I) of all
diffeomorphisms of an interval I in R. There is much classical work on the
subject, solving the conjugacy problem for special classes of maps.
Unfortunately, it is also true that many results and arguments known to the
experts are difficult to find in the literature, or simply absent. We try to
repair these lacunae, by giving a systematic review, and we also include new
results about the conjugacy classification in the general case.Comment: 53 page
Flowability of plane homeomorphisms
We describe necessary and sufficient conditions for an orientation preserving
fixed point free planar homeomorphism that preserves the standard Reeb
foliation to embed in a planar flow that leaves the foliation invariant.Comment: 20 pages, 3 figure
The spectral radius formula for Fourier-Stieltjes algebras
In this short note we first extend the validity of the spectral radius
formula obtained in \cite{ag} to Fourier--Stieltjes algebras. The second part
is devoted to showing that for the measure algebra on any locally compact
non-discrete Abelian group there are no non-trivial constraints between three
quantities: the norm, the spectral radius and the supremum of the
Fourier--Stieltjes transform even if we restrict our attention to measures with
all convolution powers singular with respect to Haar measure
Some new examples of recurrence and non-recurrence sets for products of rotations on the unit circle
summary:We study recurrence and non-recurrence sets for dynamical systems on compact spaces, in particular for products of rotations on the unit circle . A set of integers is called -Bohr if it is recurrent for all products of rotations on , and Bohr if it is recurrent for all products of rotations on . It is a result due to Katznelson that for each there exist sets of integers which are -Bohr but not -Bohr. We present new examples of -Bohr sets which are not Bohr, thanks to a construction which is both flexible and completely explicit. Our results are related to an old combinatorial problem of Veech concerning syndetic sets and the Bohr topology on , and its reformulation in terms of recurrence sets which is due to Glasner and Weiss