Expand all Collapse all | Results 1 - 21 of 21 |
1. CJM Online first
Hyperspace Dynamics of Generic Maps of the Cantor Space We study the hyperspace dynamics induced from generic continuous maps
and from generic homeomorphisms of the Cantor space, with emphasis on the
notions of Li-Yorke chaos, distributional chaos, topological entropy,
chain continuity, shadowing and recurrence.
Keywords:cantor space, continuous maps, homeomorphisms, hyperspace, dynamics Categories:37B99, 54H20, 54E52 |
2. CJM Online first
Addendum to ``Nearly Countable Dense Homogeneous Spaces' This paper provides an addendum to M. HruÅ¡Ã¡k
and J. van Mill ``Nearly countable dense homogeneous spaces.''
Canad. J. Math., published online 2013-03-08
http://dx.doi.org/10.4153/CJM-2013-006-8.
Keywords:countable dense homogeneous, nearly countable dense homogeneous, Effros Theorem, Vaught's conjecture Categories:54H05, 03E15, 54E50 |
3. CJM 2013 (vol 65 pp. 1287)
$K$-theory of Furstenberg Transformation Group $C^*$-algebras The paper studies the $K$-theoretic invariants of the crossed product
$C^{*}$-algebras associated with an important family of homeomorphisms
of the tori $\mathbb{T}^{n}$ called Furstenberg transformations.
Using the Pimsner-Voiculescu theorem, we prove that given $n$, the
$K$-groups of those crossed products, whose corresponding $n\times n$
integer matrices are unipotent of maximal degree, always have the same
rank $a_{n}$. We show using the theory developed here that a claim
made in the literature about the torsion subgroups of these $K$-groups
is false. Using the representation theory of the simple Lie algebra
$\frak{sl}(2,\mathbb{C})$, we show that, remarkably, $a_{n}$ has a
combinatorial significance. For example, every $a_{2n+1}$ is just the
number of ways that $0$ can be represented as a sum of integers
between $-n$ and $n$ (with no repetitions). By adapting an argument
of van Lint (in which he answered a question of ErdÅs), a simple,
explicit formula for the asymptotic behavior of the sequence
$\{a_{n}\}$ is given. Finally, we describe the order structure of the
$K_{0}$-groups of an important class of Furstenberg crossed products,
obtaining their complete Elliott invariant using classification
results of H. Lin and N. C. Phillips.
Keywords:$K$-theory, transformation group $C^*$-algebra, Furstenberg transformation, Anzai transformation, minimal homeomorphism, positive cone, minimal homeomorphism Categories:19K14, 19K99, 46L35, 46L80, , 05A15, 05A16, 05A17, 15A36, 17B10, 17B20, 37B05, 54H20 |
4. CJM Online first
Nearly Countable Dense Homogeneous Spaces We study separable metric spaces with few types of countable dense
sets. We present a structure theorem for locally compact spaces
having precisely $n$ types of countable dense sets: such a space
contains a subset $S$ of size at most $n{-}1$ such that $S$ is
invariant under
all homeomorphisms of $X$ and $X\setminus S$ is countable dense
homogeneous. We prove that every Borel space having fewer than $\mathfrak{c}$
types of
countable dense sets is Polish. The natural question of whether every
Polish space has either countably many or $\mathfrak{c}$ many types of
countable
dense sets, is shown to be closely related to Topological Vaught's
Conjecture.
Keywords:countable dense homogeneous, nearly countable dense homogeneous, Effros Theorem, Vaught's conjecture Categories:54H05, 03E15, 54E50 |
5. CJM 2012 (vol 64 pp. 1182)
PFA$(S)[S]$: More Mutually Consistent Topological Consequences of $PFA$ and $V=L$ Extending the work of Larson and Todorcevic,
we show there
is a model of set theory in which normal spaces are collectionwise
Hausdorff if they are either first countable or locally compact, and
yet there are no first countable $L$-spaces or compact
$S$-spaces. The model is one of the form PFA$(S)[S]$, where $S$
is a coherent Souslin tree.
Keywords:PFA$(S)[S]$, proper forcing, coherent Souslin tree, locally compact, normal, collectionwise Hausdorff, supercompact cardinal Categories:54A35, 54D15, 54D20, 54D45, 03E35, 03E57, 03E65 |
6. CJM 2011 (vol 63 pp. 533)
On Best Proximity Points in Metric and Banach Spaces In this paper we study the existence and uniqueness of
best proximity points of cyclic contractions as well as the convergence
of iterates to such proximity points. We take two different approaches,
each one leading to different results that complete, if not improve,
other similar results in the theory. Results in this paper stand for Banach
spaces, geodesic metric spaces and metric spaces. We also include an appendix
on CAT$(0)$ spaces where we study the particular behavior of these spaces
regarding the problems we are concerned with.
Categories:54H25, 47H09 |
7. CJM 2009 (vol 62 pp. 262)
On the Spectrum of the Equivariant Cohomology Ring If an algebraic torus $T$ acts on a complex projective algebraic
variety $X$, then the affine scheme $\operatorname{Spec}
H^*_T(X;\mathbb C)$ associated with the equivariant cohomology is
often an arrangement of linear subspaces of the vector space
$H_2^T(X;\mathbb C).$ In many situations the ordinary cohomology ring
of $X$ can be described in terms of this arrangement.
Categories:14L30, 54H15 |
8. CJM 2009 (vol 62 pp. 182)
Mutually Aposyndetic Decomposition of Homogeneous Continua A new decomposition, the \emph{mutually aposyndetic decomposition} of
homogeneous continua into closed, homogeneous sets is introduced. This
decomposition is respected by homeomorphisms and topologically
unique. Its quotient is a mutually aposyndetic homogeneous continuum,
and in all known examples, as well as in some general cases, the
members of the decomposition are semi-indecomposable continua. As
applications, we show that hereditarily decomposable homogeneous
continua and path connected homogeneous continua are mutually
aposyndetic. A class of new examples of homogeneous continua is
defined. The mutually aposyndetic decomposition of each of these
continua is non-trivial and different from Jones' aposyndetic
decomposition.
Keywords:ample, aposyndetic, continuum, decomposition, filament, homogeneous Categories:54F15, 54B15 |
9. CJM 2009 (vol 61 pp. 708)
Regular Homeomorphisms of Finite Order on Countable Spaces We present a structure theorem for a broad class of homeomorphisms of
finite order on countable zero dimensional spaces. As applications we
show the following.
\begin{compactenum}[\rm(a)]
\item Every countable nondiscrete topological group not containing an
open Boolean subgroup can be partitioned into infinitely many dense
subsets.
\item If $G$ is a countably infinite Abelian group with finitely many
elements of order $2$ and $\beta G$ is the Stone--\v Cech
compactification of $G$ as a discrete semigroup, then for every
idempotent $p\in\beta G\setminus\{0\}$, the subset
$\{p,-p\}\subset\beta G$ generates algebraically the free product of
one-element semigroups $\{p\}$ and~$\{-p\}$.
\end{compactenum}
Keywords:Homeomorphism, homogeneous space, topological group, resolvability, Stone-\v Cech compactification Categories:22A30, 54H11, 20M15, 54A05 |
10. CJM 2009 (vol 61 pp. 604)
First Countable Continua and Proper Forcing Assuming the Continuum Hypothesis,
there is a compact, first countable, connected space of weight $\aleph_1$
with no totally disconnected perfect subsets.
Each such space, however, may be destroyed by
some proper forcing order which does not add reals.
Keywords:connected space, Continuum Hypothesis, proper forcing, irreducible map Categories:54D05, 03E35 |
11. CJM 2009 (vol 61 pp. 124)
Characterizing Complete Erd\H os Space The space now known as {\em complete Erd\H os
space\/} $\cerdos$ was introduced by Paul Erd\H os in 1940 as the
closed subspace of the Hilbert space $\ell^2$ consisting of all
vectors such that every coordinate is in the convergent sequence
$\{0\}\cup\{1/n:n\in\N\}$. In a solution to a problem posed by Lex G.
Oversteegen we present simple and useful topological
characterizations of $\cerdos$.
As an application we determine the class
of factors of $\cerdos$. In another application we determine
precisely which of the spaces that can be constructed in the Banach
spaces $\ell^p$ according to the `Erd\H os method' are homeomorphic
to $\cerdos$. A novel application states that if $I$ is a
Polishable $F_\sigma$-ideal on $\omega$, then $I$ with the Polish
topology is homeomorphic to either $\Z$, the Cantor set $2^\omega$,
$\Z\times2^\omega$, or $\cerdos$. This last result answers a
question that was asked
by Stevo Todor{\v{c}}evi{\'c}.
Keywords:Complete Erd\H os space, Lelek fan, almost zero-dimensional, nowhere zero-dimensional, Polishable ideals, submeasures on $\omega$, $\R$-trees, line-free groups in Banach spaces Categories:28C10, 46B20, 54F65 |
12. CJM 2008 (vol 60 pp. 1149)
Conjugate Reciprocal Polynomials with All Roots on the Unit Circle We study the geometry, topology and Lebesgue measure of the set of
monic conjugate reciprocal polynomials of fixed degree with all
roots on the unit circle. The set of such polynomials of degree $N$
is naturally associated to a subset of $\R^{N-1}$. We calculate
the volume of this set, prove the set is homeomorphic to the $N-1$
ball and that its isometry group is isomorphic to the dihedral
group of order $2N$.
Categories:11C08, 28A75, 15A52, 54H10, 58D19 |
13. CJM 2007 (vol 59 pp. 1008)
Ideas from Zariski Topology in the Study of Cubical Homology Cubical sets and their homology have been
used in dynamical systems as well as in digital imaging. We take a
fresh look at this topic, following Zariski ideas from
algebraic geometry. The cubical topology is defined to be a
topology in $\R^d$ in which a set is closed if and only if it is
cubical. This concept is a convenient frame for describing a
variety of important features of cubical sets. Separation axioms
which, in general, are not satisfied here, characterize exactly
those pairs of points which we want to distinguish. The noetherian
property guarantees the correctness of the algorithms. Moreover, maps
between cubical sets which are continuous and closed with respect
to the cubical topology are precisely those for whom the homology
map can be defined and computed without grid subdivisions. A
combinatorial version of the Vietoris-Begle theorem is derived. This theorem
plays the central role in an algorithm computing homology
of maps which are continuous
with respect to the Euclidean topology.
Categories:55-04, 52B05, 54C60, 68W05, 68W30, 68U10 |
14. CJM 2007 (vol 59 pp. 465)
Searching for Absolute $\mathcal{CR}$-Epic Spaces In previous papers, Barr and Raphael investigated the situation of a
topological space $Y$ and a subspace $X$ such that the induced map
$C(Y)\to C(X)$ is an epimorphism in the category $\CR$ of commutative
rings (with units). We call such an embedding a $\CR$-epic embedding
and we say that $X$ is absolute $\CR$-epic if every embedding of $X$
is $\CR$-epic. We continue this investigation. Our most notable
result shows that a Lindel\"of space $X$ is absolute $\CR$-epic if a
countable intersection of $\beta X$-neighbourhoods of $X$ is a $\beta
X$-neighbourhood of $X$. This condition is stable under countable
sums, the formation of closed subspaces, cozero-subspaces, and being
the domain or codomain of a perfect map. A strengthening of the
Lindel\"of property leads to a new class with the same closure
properties that is also closed under finite products. Moreover, all
\s-compact spaces and all Lindel\"of $P$-spaces satisfy this stronger
condition. We get some results in the non-Lindel\"of case that are
sufficient to show that the Dieudonn\'e plank and some closely related
spaces are absolute $\CR$-epic.
Keywords:absolute $\mathcal{CR}$-epics, countable neighbourhoo9d property, amply LindelÃ¶f, DiuedonnÃ© plank Categories:18A20, 54C45, 54B30 |
15. CJM 2005 (vol 57 pp. 1121)
On $\mathcal{CR}$-epic Embeddings and Absolute $\mathcal{CR}$-epic Spaces We study Tychonoff spaces $X$ with the property that, for all
topological embeddings $X\to Y $, the induced map $C(Y) \to C(X)$ is an
epimorphism of rings. Such spaces are called \good. The simplest
examples of \good spaces are $\sigma$-compact locally compact spaces and
\Lin $P$-spaces. We show that \good first countable spaces must be
locally compact.
However, a ``bad'' class of \good spaces is exhibited whose pathology
settles, in the negative, a number of open questions. Spaces which are
not \good abound, and some are presented.
Categories:18A20, 54C45, 54B30 |
16. CJM 2004 (vol 56 pp. 825)
Differentiability Properties of Optimal Value Functions Differentiability properties of optimal value functions associated with
perturbed optimization problems require strong assumptions. We consider such
a set of assumptions which does not use compactness hypothesis but which
involves a kind of coherence property. Moreover, a strict differentiability
property is obtained by using techniques of Ekeland and Lebourg and a result
of Preiss. Such a strengthening is required in order to obtain genericity
results.
Keywords:differentiability, generic, marginal, performance function, subdifferential Categories:26B05, 65K10, 54C60, 90C26, 90C48 |
17. CJM 2002 (vol 54 pp. 1187)
On the Injectivity of $C^1$ Maps of the Real Plane Let $X\colon\mathbb{R}^2\to\mathbb{R}^2$ be a $C^1$ map. Denote by $\Spec(X)$ the set of
(complex) eigenvalues of $\DX_p$ when $p$ varies in $\mathbb{R}^2$. If there exists
$\epsilon >0$ such that $\Spec(X)\cap(-\epsilon,\epsilon)=\emptyset$, then
$X$ is injective. Some applications of this result to the real Keller Jacobian
conjecture are discussed.
Categories:34D05, 54H20, 58F10, 58F21 |
18. CJM 2001 (vol 53 pp. 325)
Ext and OrderExt Classes of Certain Automorphisms of $C^*$-Algebras Arising from Cantor Minimal Systems |
Ext and OrderExt Classes of Certain Automorphisms of $C^*$-Algebras Arising from Cantor Minimal Systems Giordano, Putnam and Skau showed that the transformation group
$C^*$-algebra arising from a Cantor minimal system is an $AT$-algebra,
and classified it by its $K$-theory. For approximately inner
automorphisms that preserve $C(X)$, we will determine their classes in
the Ext and OrderExt groups, and introduce a new invariant for the
closure of the topological full group. We will also prove that every
automorphism in the kernel of the homomorphism into the Ext group is
homotopic to an inner automorphism, which extends Kishimoto's result.
Categories:46L40, 46L80, 54H20 |
19. CJM 1999 (vol 51 pp. 309)
Symmetric sequence subspaces of $C(\alpha)$, II If $\alpha$ is an ordinal, then the space of all ordinals less than or
equal to $\alpha$ is a compact Hausdorff space when endowed with the
order topology. Let $C(\alpha)$ be the space of all continuous
real-valued functions defined on the ordinal interval $[0,
\alpha]$. We characterize the symmetric sequence spaces which embed
into $C(\alpha)$ for some countable ordinal $\alpha$. A hierarchy
$(E_\alpha)$ of symmetric sequence spaces is constructed so that, for
each countable ordinal $\alpha$, $E_\alpha$ embeds into
$C(\omega^{\omega^\alpha})$, but does not embed into
$C(\omega^{\omega^\beta})$ for any $\beta < \alpha$.
Categories:03E13, 03E15, 46B03, 46B45, 46E15, 54G12 |
20. CJM 1998 (vol 50 pp. 342)
Shape fibrations, multivalued maps and shape groups The notion of shape fibration with the near lifting of near
multivalued paths property is studied. The relation of these
maps---which agree with shape fibrations having totally disconnected
fibers---with Hurewicz fibrations with the unique path lifting
property is completely settled. Some results concerning homotopy and
shape groups are presented for shape fibrations with the near lifting
of near multivalued paths property. It is shown that for this class of
shape fibrations the existence of liftings of a fine multivalued map,
is equivalent to an algebraic problem relative to the homotopy, shape
or strong shape groups associated.
Keywords:Shape fibration, multivalued map, homotopy groups, shape, groups, strong shape groups Categories:54C56, 55P55, 55Q05, 55Q07, 55R05 |
21. CJM 1997 (vol 49 pp. 1089)
Sets on which measurable functions are determined by their range We study sets on which measurable real-valued functions on a
measurable space with negligibles are determined by their range.
Keywords:measurable function, measurable space with negligibles, continuous image, set of range uniqueness (SRU) Categories:28A20, 28A05, 54C05, 26A30, 03E35, 03E50 |