508 research outputs found
On periodic homeomorphisms of spheres
The purpose of this paper is to study how small orbits of periodic
homemorphisms of spheres can be.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-22.abs.htm
Connectedness properties of the set where the iterates of an entire function are bounded
We investigate some connectedness properties of the set of points K(f) where
the iterates of an entire function f are bounded. In particular, we describe a
class of transcendental entire functions for which an analogue of the
Branner-Hubbard conjecture holds and show that, for such functions, if K(f) is
disconnected then it has uncountably many components. We give examples to show
that K(f) can be totally disconnected, and we use quasiconformal surgery to
construct a function for which K(f) has a component with empty interior that is
not a singleton.Comment: 21 page
Mean Dimension & Jaworski-type Theorems
According to the celebrated Jaworski Theorem, a finite dimensional aperiodic
dynamical system embeds in the -dimensional cubical shift
. If admits periodic points (still assuming
) then we show in this paper that periodic dimension
implies that embeds in the -dimensional
cubical shift . This verifies a conjecture by
Lindenstrauss and Tsukamoto for finite dimensional systems. Moreover for an
infinite dimensional dynamical system, with the same periodic dimension
assumption, the set of periodic points can be equivariantly immersed in
. Furthermore we introduce a notion of
markers for general topological dynamical systems, and use a generalized
version of the Bonatti-Crovisier tower theorem, to show that an extension
of an aperiodic finite-dimensional system whose mean dimension obeys
embeds in the -cubical shift
.Comment: To appear in Proceedings of the London Mathematical Societ
The combinatorics of Borel covers
In this paper we extend previous studies of selection principles for families
of open covers of sets of real numbers to also include families of countable
Borel covers.
The main results of the paper could be summarized as follows:
1. Some of the classes which were different for open covers are equal for
Borel covers -- Section 1;
2. Some Borel classes coincide with classes that have been studied under a
different guise by other authors -- Section 4.Comment: Regular updat
Combination of convergence groups
We state and prove a combination theorem for relatively hyperbolic groups
seen as geometrically finite convergence groups. For that, we explain how to
contruct a boundary for a group that is an acylindrical amalgamation of
relatively hyperbolic groups over a fully quasi-convex subgroup. We apply our
result to Sela's theory on limit groups and prove their relative hyperbolicity.
We also get a proof of the Howson property for limit groups.Comment: Published in Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol7/paper27.abs.htm
Covering dimension and finite-to-one maps
Hurewicz' characterized the dimension of separable metrizable spaces by means
of finite-to-one maps. We investigate whether this characterization also holds
in the class of compact F-spaces of weight c. Our main result is that, assuming
the Continuum Hypothesis, an n-dimensional compact F-space of weight c is the
continuous image of a zero-dimensional compact Hausdorff space by an at most
2n-to-1 map
On the Chabauty space of locally compact abelian groups
This paper contains several results about the Chabauty space of a general
locally compact abelian group. Notably, we determine its topological dimension,
we characterize when it is totally disconnected or connected; we characterize
isolated points.Comment: 24 pages, 0 figur
On disjoint Borel uniformizations
Larman showed that any closed subset of the plane with uncountable vertical
cross-sections has aleph_1 disjoint Borel uniformizing sets. Here we show that
Larman's result is best possible: there exist closed sets with uncountable
cross-sections which do not have more than aleph_1 disjoint Borel
uniformizations, even if the continuum is much larger than aleph_1. This
negatively answers some questions of Mauldin. The proof is based on a result of
Stern, stating that certain Borel sets cannot be written as a small union of
low-level Borel sets. The proof of the latter result uses Steel's method of
forcing with tagged trees; a full presentation of this method, written in terms
of Baire category rather than forcing, is given here
Mean dimension and a sharp embedding theorem: extensions of aperiodic subshifts
We show that if is an extension of an aperiodic subshift (a subsystem
of for some ) and
has mean dimension ), then it embeds
equivariantly in (([0,1]^{D})^{\mathbb{Z}},\mathrm{shift})(X,T)(([0,1]^{D+1})^{\mathbb{Z}},\mathrm{shift})$
Topological classification of affine operators on unitary and Euclidean spaces
We classify affine operators on a unitary or Euclidean space U up to
topological conjugacy. An affine operator is a map f: U-->U of the form
f(x)=Ax+b, in which A: U-->U is a linear operator and b in U. Two affine
operators f and g are said to be topologically conjugate if hg=fh for some
homeomorphism h: U-->U
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