Expand all Collapse all | Results 1 - 24 of 24 |
1. CMB Online first
Simplicity of Partial Skew Group Rings of Abelian Groups Let $A$ be a ring with local units, $E$ a set of local units for $A$,
$G$ an abelian group and $\alpha$ a partial action of $G$ by ideals of
$A$ that contain local units.
We show that $A\star_{\alpha} G$ is simple if and only if $A$ is
$G$-simple and the center of the corner $e\delta_0 (A\star_{\alpha} G)
e \delta_0$ is a field for all $e\in E$. We apply the result to
characterize simplicity of partial skew group rings in two cases,
namely for partial skew group rings arising from partial actions by
clopen subsets of a compact set and partial actions on the set level.
Keywords:partial skew group rings, simple rings, partial actions, abelian groups Categories:16S35, 37B05 |
2. CMB 2012 (vol 57 pp. 240)
Addendum to ``Limit Sets of Typical Homeomorphisms'' Given an integer $n \geq 3$,
a metrizable compact topological $n$-manifold $X$ with boundary,
and a finite positive Borel measure $\mu$ on $X$,
we prove that for the typical homeomorphism $f : X \to X$,
it is true that for $\mu$-almost every point $x$ in $X$ the restriction of
$f$ (respectively of $f^{-1}$) to the omega limit set $\omega(f,x)$
(respectively to the alpha limit set $\alpha(f,x)$) is topologically
conjugate to the universal odometer.
Keywords:topological manifolds, homeomorphisms, measures, Baire category, limit sets Categories:37B20, 54H20, 28C15, 54C35, 54E52 |
3. CMB 2012 (vol 56 pp. 709)
Universal Minimal Flows of Groups of Automorphisms of Uncountable Structures It is a well-known fact, that the greatest ambit for
a topological group $G$ is the Samuel compactification of $G$ with
respect to the right uniformity on $G.$ We apply the original
description by Samuel from 1948 to give a simple computation of the
universal minimal flow for groups of automorphisms of uncountable
structures using FraÃ¯ssÃ© theory and Ramsey theory. This work
generalizes some of the known results about countable structures.
Keywords:universal minimal flows, ultrafilter flows, Ramsey theory Categories:37B05, 03E02, 05D10, 22F50, 54H20 |
4. CMB 2012 (vol 56 pp. 477)
Hypercyclic Abelian Groups of Affine Maps on $\mathbb{C}^{n}$ We give a characterization of hypercyclic abelian group
$\mathcal{G}$ of affine maps on $\mathbb{C}^{n}$. If $\mathcal{G}$
is finitely generated, this characterization is explicit. We prove
in particular
that no abelian group generated by $n$ affine maps on $\mathbb{C}^{n}$ has a dense orbit.
Keywords:affine, hypercyclic, dense, orbit, affine group, abelian Categories:37C85, 47A16 |
5. CMB 2011 (vol 56 pp. 136)
On Constructing Ergodic Hyperfinite Equivalence Relations of Non-Product Type Product type equivalence relations are hyperfinite measured
equivalence relations, which, up to orbit equivalence, are generated
by product type odometer actions. We give a concrete example of a
hyperfinite equivalence relation of non-product type, which is the
tail equivalence on a Bratteli diagram.
In order to show that the equivalence relation constructed is not of
product type we will use a criterion called property A. This
property, introduced by Krieger for non-singular transformations, is
defined directly for hyperfinite equivalence relations in this paper.
Keywords:property A, hyperfinite equivalence relation, non-product type Categories:37A20, 37A35, 46L10 |
6. CMB 2011 (vol 55 pp. 858)
An Optimal Transport View of SchrÃ¶dinger's Equation We show that the SchrÃ¶dinger equation is a lift of Newton's third law
of motion $\nabla^\mathcal W_{\dot \mu} \dot \mu = -\nabla^\mathcal W F(\mu)$ on
the space of probability measures, where derivatives are taken
with respect to the Wasserstein Riemannian metric. Here the potential
$\mu \to F(\mu)$ is the sum of the total classical potential energy $\langle V,\mu\rangle$
of the extended system
and its Fisher information
$ \frac {\hbar^2} 8 \int |\nabla \ln \mu |^2
\,d\mu$. The precise relation is established via a well-known
(Madelung) transform which is shown to be a symplectic submersion
of the standard symplectic
structure of complex valued functions into the
canonical symplectic space over the Wasserstein space.
All computations are conducted in the framework of Otto's formal
Riemannian calculus for optimal transportation of probability
measures.
Keywords:SchrÃ¶dinger equation, optimal transport, Newton's law, symplectic submersion Categories:81C25, 82C70, 37K05 |
7. CMB 2011 (vol 55 pp. 708)
Improved Range in the Return Times Theorem We prove that the Return Times Theorem holds true for pairs of $L^p-L^q$ functions,
whenever $\frac{1}{p}+\frac{1}{q}<\frac{3}{2}$.
Keywords:Return Times Theorem, maximal multiplier, maximal inequality Categories:42B25, 37A45 |
8. CMB 2011 (vol 55 pp. 297)
The Group $\operatorname{Aut}(\mu)$ is Roelcke Precompact Following a similar result of Uspenskij on the unitary group of a
separable Hilbert space, we show that, with respect to the lower (or
Roelcke) uniform structure, the Polish group $G=
\operatorname{Aut}(\mu)$ of automorphisms of an atomless standard
Borel probability space $(X,\mu)$ is precompact. We identify the
corresponding compactification as the space of Markov operators on
$L_2(\mu)$ and deduce that the algebra of right and left uniformly
continuous functions, the algebra of weakly almost periodic functions,
and the algebra of Hilbert functions on $G$, i.e., functions on
$G$ arising from unitary representations, all coincide. Again
following Uspenskij, we also conclude that $G$ is totally minimal.
Keywords:Roelcke precompact, unitary group, measure preserving transformations, Markov operators, weakly almost periodic functions Categories:54H11, 22A05, 37B05, 54H20 |
9. CMB 2011 (vol 55 pp. 225)
Limit Sets of Typical Homeomorphisms Given an integer $n \geq 3$, a metrizable compact
topological $n$-manifold $X$ with boundary, and a finite positive Borel
measure $\mu$ on $X$, we prove that for the typical homeomorphism
$f \colon X \to X$, it is true that for $\mu$-almost every point $x$ in $X$
the limit set $\omega(f,x)$ is a Cantor set of Hausdorff dimension zero,
each point of $\omega(f,x)$ has a dense orbit in $\omega(f,x)$, $f$ is
non-sensitive at each point of $\omega(f,x)$, and the function
$a \to \omega(f,a)$ is continuous at $x$.
Keywords:topological manifolds, homeomorphisms, measures, Baire category, limit sets Categories:37B20, 54H20, 28C15, 54C35, 54E52 |
10. CMB 2011 (vol 54 pp. 676)
Quasi-isometry and Plaque Expansiveness We show that a partially hyperbolic diffeomorphism is plaque
expansive (a form of structural stability for its center foliation) if the
strong stable and unstable foliations are quasi-isometric in the universal
cover. In particular, all partially hyperbolic diffeomorphisms on the 3-torus
are plaque expansive.
Category:37D30 |
11. CMB 2011 (vol 54 pp. 311)
Some Remarks Concerning the Topological Characterization of Limit Sets for Surface Flows We give some extension to theorems of JimÃ©nez LÃ³pez and Soler LÃ³pez concerning the topological characterization for limit sets of continuous flows on closed orientable surfaces.
Keywords:flows on surfaces, orbits, class of an orbit, singularities, minimal set, limit set, regular cylinder Categories:37B20, 37E35 |
12. CMB 2009 (vol 53 pp. 295)
The Global Attractor of a Damped, Forced Hirota Equation in $H^1$ The existence of the global attractor of a damped
forced Hirota equation in the phase space $H^1(\mathbb R)$ is proved. The
main idea is to establish the so-called asymptotic compactness
property of the solution operator by energy equation approach.
Keywords:global attractor, Fourier restriction norm, damping system, asymptotic compactness Categories:35Q53, 35B40, 35B41, 37L30 |
13. CMB 2008 (vol 51 pp. 545)
$C^{\ast}$-Algebras Associated with Mauldin--Williams Graphs A Mauldin--Williams graph $\mathcal{M}$ is a generalization of an
iterated function system by a directed graph. Its invariant set $K$
plays the role of the self-similar set. We associate a $C^{*}$-algebra
$\mathcal{O}_{\mathcal{M}}(K)$ with a Mauldin--Williams graph $\mathcal{M}$
and the invariant set $K$, laying emphasis on the singular points.
We assume that the underlying graph $G$ has no sinks and no sources.
If $\mathcal{M}$ satisfies the open set condition in $K$, and $G$
is irreducible and is not a cyclic permutation, then the associated
$C^{*}$-algebra $\mathcal{O}_{\mathcal{M}}(K)$ is simple and purely
infinite. We calculate the $K$-groups for some examples including the
inflation rule of the Penrose tilings.
Categories:46L35, 46L08, 46L80, 37B10 |
14. CMB 2007 (vol 50 pp. 418)
A Short Proof of Affability for Certain Cantor Minimal $\Z^2$-Systems We will show that any extension of a product of two Cantor minimal
$\Z$-systems is affable in the sense of Giordano, Putnam and Skau.
Category:37B05 |
15. CMB 2006 (vol 49 pp. 203)
The Ergodic Hilbert Transform for Admissible Processes It is shown that the ergodic Hilbert transform
exists for a class of bounded symmetric admissible processes
relative to invertible measure preserving transformations. This
generalizes the well-known result on the existence of the ergodic
Hilbert transform.
Keywords:Hilbert transform, admissible processes Categories:28D05, 37A99 |
16. CMB 2005 (vol 48 pp. 302)
Discrete Sets and Associated Dynamical\\ Systems in a Non-Commutative Setting We define a uniform structure on the set of discrete sets of a locally
compact topological space on which a locally compact topological group
acts continuously. Then we investigate the completeness of these
uniform spaces and study these spaces by means of topological
dynamical systems.
Categories:52C23, 37B50 |
17. CMB 2005 (vol 48 pp. 3)
Quantum Ergodicity of Boundary Values of Eigenfunctions: A Control Theory Approach Consider $M$, a bounded domain in ${\mathbb R}^d$, which is a
Riemanian manifold with piecewise smooth boundary and suppose that the
billiard associated to the geodesic flow reflecting on the boundary
according to the laws of geometric optics is ergodic.
We prove that the boundary value of the eigenfunctions of the Laplace
operator with reasonable boundary conditions are asymptotically
equidistributed in the boundary, extending previous results by
G\'erard and Leichtnam as well as Hassell and Zelditch,
obtained under the additional assumption of the convexity of~$M$.
Categories:35Q55, 35BXX, 37K05, 37L50, 81Q20 |
18. CMB 2004 (vol 47 pp. 553)
A Geometric Approach to Voiculescu-Brown Entropy A basic problem in dynamics is to identify systems
with positive entropy, i.e., systems which are ``chaotic.'' While
there is a vast collection of results addressing this issue in
topological dynamics, the phenomenon of positive entropy remains by and
large a mystery within the broader noncommutative domain of $C^*$-algebraic
dynamics. To shed some light on the noncommutative situation we propose
a geometric perspective inspired by work of Glasner and Weiss on
topological entropy.
This is a written version of the author's talk
at the Winter 2002 Meeting of the Canadian Mathematical Society
in Ottawa, Ontario.
Categories:46L55, 37B40 |
19. CMB 2004 (vol 47 pp. 332)
Recurrent Geodesics in Flat Lorentz $3$-Manifolds Let $M$ be a complete flat Lorentz $3$-manifold $M$ with purely
hyperbolic holonomy $\Gamma$. Recurrent geodesic rays are completely
classified when $\Gamma$ is cyclic. This implies that for any pair of
periodic geodesics $\gamma_1$, $\gamma_2$, a unique geodesic forward
spirals towards $\gamma_1$ and backward spirals towards $\gamma_2$.
Keywords:geometric structures on low-dimensional manifolds, notions of recurrence Categories:57M50, 37B20 |
20. CMB 2004 (vol 47 pp. 168)
Kolakoski-$(3,1)$ Is a (Deformed) Model Set Unlike the (classical) Kolakoski sequence on the alphabet $\{1,2\}$, its analogue
on $\{1,3\}$ can be related to a primitive substitution rule. Using this connection,
we prove that the corresponding bi-infinite fixed point is a regular generic model set
and thus has a pure point diffraction spectrum. The Kolakoski-$(3,1)$ sequence is
then obtained as a deformation, without losing the pure point diffraction property.
Categories:52C23, 37B10, 28A80, 43A25 |
21. CMB 2003 (vol 46 pp. 277)
Rigidity of Hamiltonian Actions This paper studies the following question: Given an
$\omega'$-symplectic action of a Lie group on a manifold $M$ which
coincides, as a smooth action, with a Hamiltonian $\omega$-action,
when is this action a Hamiltonian $\omega'$-action? Using a result of
Morse-Bott theory presented in Section~2, we show in Section~3 of this
paper that such an action is in fact a Hamiltonian $\omega'$-action,
provided that $M$ is compact and that the Lie group is compact and
connected. This result was first proved by Lalonde-McDuff-Polterovich
in 1999 as a consequence of a more general theory that made use of
hard geometric analysis. In this paper, we prove it using classical
methods only.
Categories:53D05, 37J25 |
22. CMB 2002 (vol 45 pp. 697)
Pure Discrete Spectrum for One-dimensional Substitution Systems of Pisot Type We consider two dynamical systems associated with a substitution of
Pisot type: the usual $\mathbb{Z}$-action on a sequence space, and
the $\mathbb{R}$-action, which can be defined as a tiling dynamical
system or as a suspension flow. We describe procedures for checking
when these systems have pure discrete spectrum (the ``balanced
pairs algorithm'' and the ``overlap algorithm'') and study the
relation between them. In particular, we show that pure discrete
spectrum for the $\mathbb{R}$-action implies pure discrete spectrum
for the $\mathbb{Z}$-action, and obtain a partial result in the
other direction. As a corollary, we prove pure discrete spectrum
for every $\mathbb{R}$-action associated with a two-symbol
substitution of Pisot type (this is conjectured for an arbitrary
number of symbols).
Categories:37A30, 52C23, 37B10 |
23. CMB 2002 (vol 45 pp. 123)
Uniform Distribution in Model Sets We give a new measure-theoretical proof of the uniform distribution
property of points in model sets (cut and project sets). Each model
set comes as a member of a family of related model sets, obtained by
joint translation in its ambient (the `physical') space and its
internal space. We prove, assuming only that the window defining the
model set is measurable with compact closure, that almost surely the
distribution of points in any model set from such a family is uniform
in the sense of Weyl, and almost surely the model set is pure point
diffractive.
Categories:52C23, 11K70, 28D05, 37A30 |
24. CMB 2001 (vol 44 pp. 292)
An Analogue of Napoleon's Theorem in the Hyperbolic Plane There is a theorem, usually attributed to Napoleon, which states that
if one takes any triangle in the Euclidean Plane, constructs
equilateral triangles on each of its sides, and connects the midpoints
of the three equilateral triangles, one will obtain an equilateral
triangle. We consider an analogue of this problem in the hyperbolic
plane.
Category:37D40 |