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1. CJM Online first
Minimal Dynamical Systems on Connected Odd Dimensional Spaces Let $\beta\colon S^{2n+1}\to S^{2n+1}$ be a minimal homeomorphism ($n\ge 1$). We show that
the crossed product $C(S^{2n+1})\rtimes_\beta \mathbb{Z}$ has rational tracial rank at most one.
Let $\Omega$ be a connected compact metric space with finite covering dimension and
with $H^1(\Omega, \mathbb{Z})=\{0\}.$ Suppose that $K_i(C(\Omega))=\mathbb{Z}\oplus G_i,$ where $G_i$ is a finite abelian group, $i=0,1.$
Let $\beta\colon \Omega\to\Omega$ be a minimal homeomorphism. We also show that
$A=C(\Omega)\rtimes_\beta\mathbb{Z}$ has rational tracial rank at most one and is
classifiable.
In particular, this applies to the minimal dynamical systems on
odd dimensional real projective spaces.
This is done by studying minimal homeomorphisms on $X\times \Omega,$ where
$X$ is the Cantor set.
Keywords:minimal dynamical systems Categories:46L35, 46L05 |
2. CJM 2014 (vol 67 pp. 369)
A Free Product Formula for the Sofic Dimension It is proved that if $G=G_1*_{G_3}G_2$ is free product of probability
measure preserving $s$-regular ergodic discrete groupoids amalgamated
over an amenable subgroupoid $G_3$, then the sofic dimension $s(G)$
satisfies the equality
\[
s(G)=\mathfrak{h}(G_1^0)s(G_1)+\mathfrak{h}(G_2^0)s(G_2)-\mathfrak{h}(G_3^0)s(G_3)
\]
where $\mathfrak{h}$ is the normalized Haar measure on $G$.
Keywords:sofic groups, dynamical systems, orbit equivalence, free entropy Category:20E06 |
3. CJM Online first
A Finite-time Condition for Exponential Trichotomy in Infinite Dynamical Systems In this article we study exponential trichotomy for infinite dimensional
discrete time dynamical systems. The goal of this article is to prove that
finite time exponential trichotomy conditions allow to derive exponential
trichotomy for any times. We present an application to the case of pseudo
orbits in some neighborhood of a normally hyperbolic set.
Keywords:exponential trichotomy, exponential dichotomy, discrete time dynamical systems, difference equations Categories:34D09, 34A10 |
4. CJM 2013 (vol 65 pp. 1384)
Estimates of Hausdorff Dimension for Non-wandering Sets of Higher Dimensional Open Billiards This article concerns a class of open billiards consisting of a finite
number of strictly convex, non-eclipsing obstacles $K$. The
non-wandering set $M_0$ of the billiard ball map is a topological
Cantor set and its Hausdorff dimension has been previously estimated
for billiards in $\mathbb{R}^2$, using well-known techniques. We
extend these estimates to billiards in $\mathbb{R}^n$, and make
various refinements to the estimates. These refinements also allow
improvements to other results. We also show that in many cases, the
non-wandering set is confined to a particular subset of $\mathbb{R}^n$
formed by the convex hull of points determined by period 2
orbits. This allows more accurate bounds on the constants used in
estimating Hausdorff dimension.
Keywords:dynamical systems, billiards, dimension, Hausdorff Categories:37D20, 37D40 |
5. CJM 2012 (vol 65 pp. 349)
Ergodic Properties of Randomly Coloured Point Sets We provide a framework for studying randomly coloured point sets in a
locally compact, second-countable space on which a
metrisable unimodular group acts continuously and properly.
We first construct and describe an
appropriate dynamical system for uniformly discrete uncoloured point sets. For
point sets of finite local complexity, we
characterise ergodicity geometrically in terms of pattern frequencies.
The general framework allows to incorporate a random
colouring of the point sets. We derive an ergodic theorem for randomly
coloured point sets with finite-range dependencies.
Special attention is paid to the exclusion of exceptional instances for uniquely ergodic
systems. The setup allows for a straightforward application to randomly
coloured graphs.
Keywords:Delone sets, dynamical systems Categories:37B50, 37A30 |
6. CJM 2011 (vol 65 pp. 149)
Equicontinuous Delone Dynamical Systems We characterize equicontinuous Delone dynamical systems as those
coming from Delone sets with strongly almost periodic Dirac combs.
Within the class of systems with finite local complexity, the only
equicontinuous systems are then shown to be the crystallographic
ones. On the other hand, within the class without finite local
complexity, we exhibit examples of equicontinuous minimal Delone
dynamical systems that are not crystallographic.
Our results solve the problem posed by Lagarias as to whether a Delone
set whose Dirac comb is strongly almost periodic must be
crystallographic.
Keywords:Delone sets, tilings, diffraction, topological dynamical systems, almost periodic systems Category:37B50 |
7. CJM 2003 (vol 55 pp. 3)
An Exactly Solved Model for Mutation, Recombination and Selection It is well known that rather general mutation-recombination models can be
solved algorithmically (though not in closed form) by means of Haldane
linearization. The price to be paid is that one has to work with a
multiple tensor product of the state space one started from.
Here, we present a relevant subclass of such models, in continuous time,
with independent mutation events at the sites, and crossover events
between them. It admits a closed solution of the corresponding
differential equation on the basis of the original state space, and
also closed expressions for the linkage disequilibria, derived by means
of M\"obius inversion. As an extra benefit, the approach can be extended
to a model with selection of additive type across sites. We also derive
a necessary and sufficient criterion for the mean fitness to be a Lyapunov
function and determine the asymptotic behaviour of the solutions.
Keywords:population genetics, recombination, nonlinear $\ODE$s, measure-valued dynamical systems, MÃ¶bius inversion Categories:92D10, 34L30, 37N30, 06A07, 60J25 |