26. CJM 2006 (vol 58 pp. 39)
 Exel, R.; Vershik, A.

$C^*$Algebras of Irreversible Dynamical Systems
We show that certain $C^*$algebras which have been studied by,
among others, Arzumanian, Vershik, Deaconu, and Renault, in
connection with a measurepreserving transformation of a measure space
or a covering map of a compact space, are special cases of the
endomorphism crossedproduct construction recently introduced by the
first named author. As a consequence these algebras are given
presentations in terms of generators and relations. These results
come as a consequence of a general theorem on faithfulness of
representations which are covariant with respect to certain circle
actions. For the case of topologically free covering maps we prove a
stronger result on faithfulness of representations which needs no
covariance. We also give a necessary and sufficient condition for
simplicity.
Categories:46L55, 37A55 

27. CJM 2005 (vol 57 pp. 1291)
 Riveros, Carlos M. C.; Tenenblat, Keti

Dupin Hypersurfaces in $\mathbb R^5$
We study Dupin
hypersurfaces in $\mathbb R^5$ parametrized by lines of curvature, with
four distinct principal curvatures. We characterize locally a generic
family of such hypersurfaces in terms of the principal curvatures and
four vector valued functions of one variable. We show that these vector
valued functions are invariant by inversions and homotheties.
Categories:53B25, 53C42, 35N10, 37K10 

28. CJM 2004 (vol 56 pp. 553)
 Mohammadalikhani, Ramin

Cohomology Ring of Symplectic Quotients by Circle Actions
In this article we are concerned with how to compute the cohomology ring
of a symplectic quotient by a circle action using the information we have
about the cohomology of the original manifold and some data at the fixed
point set of the action. Our method is based on the TolmanWeitsman theorem
which gives a characterization of the kernel of the Kirwan map. First we
compute a generating set for the kernel of the Kirwan map for the case of
product of compact connected manifolds such that the cohomology ring of each
of them is generated by a degree two class. We assume the fixed point set is
isolated; however the circle action only needs to be ``formally Hamiltonian''.
By identifying the kernel, we obtain the cohomology ring of the symplectic
quotient. Next we apply this result to some special cases and in particular
to the case of products of two dimensional spheres. We show that the results
of Kalkman and HausmannKnutson are special cases of our result.
Categories:53D20, 53D30, 37J10, 37J15, 53D05 

29. CJM 2004 (vol 56 pp. 449)
30. CJM 2004 (vol 56 pp. 115)
 Kenny, Robert

Estimates of Hausdorff Dimension for the NonWandering Set of an Open Planar Billiard
The billiard flow in the plane has a simple geometric definition; the
movement along straight lines of points except where elastic
reflections are made with the boundary of the billiard domain. We
consider a class of open billiards, where the billiard domain is
unbounded, and the boundary is that of a finite number of strictly
convex obstacles. We estimate the Hausdorff dimension of the
nonwandering set $M_0$ of the discrete time billiard ball map, which
is known to be a Cantor set and the largest invariant set. Under
certain conditions on the obstacles, we use a wellknown coding of
$M_0$ \cite{Morita} and estimates using convex fronts related to the
derivative of the billiard ball map \cite{StAsy} to estimate the
Hausdorff dimension of local unstable sets. Consideration of the
local product structure then yields the desired estimates, which
provide asymptotic bounds on the Hausdorff dimension's convergence to
zero as the obstacles are separated.
Categories:37D50, 37C45;, 28A78 

31. CJM 2003 (vol 55 pp. 247)
32. CJM 2003 (vol 55 pp. 3)
 Baake, Michael; Baake, Ellen

An Exactly Solved Model for Mutation, Recombination and Selection
It is well known that rather general mutationrecombination 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, measurevalued dynamical systems, MÃ¶bius inversion Categories:92D10, 34L30, 37N30, 06A07, 60J25 

33. CJM 2002 (vol 54 pp. 897)
 Fortuny Ayuso, Pedro

The Valuative Theory of Foliations
This paper gives a characterization of valuations that follow the
singular infinitely near points of plane vector fields, using the
notion of L'H\^opital valuation, which generalizes a well known classical
condition. With that tool, we give a valuative description of vector
fields with infinite solutions, singularities with rational quotient
of eigenvalues in its linear part, and polynomial vector fields with
transcendental solutions, among other results.
Categories:12J20, 13F30, 16W60, 37F75, 34M25 

34. CJM 2001 (vol 53 pp. 715)
 Cushman, Richard; Śniatycki, Jędrzej

Differential Structure of Orbit Spaces
We present a new approach to singular reduction of Hamiltonian systems
with symmetries. The tools we use are the category of differential
spaces of Sikorski and the StefanSussmann theorem. The former is
applied to analyze the differential structure of the spaces involved
and the latter is used to prove that some of these spaces are smooth
manifolds.
Our main result is the identification of accessible sets of the
generalized distribution spanned by the Hamiltonian vector fields of
invariant functions with singular reduced spaces. We are also able
to describe the differential structure of a singular reduced space
corresponding to a coadjoint orbit which need not be locally closed.
Keywords:accessible sets, differential space, Poisson algebra, proper action, singular reduction, symplectic manifolds Categories:37J15, 58A40, 58D19, 70H33 

35. CJM 2001 (vol 53 pp. 382)
 Pivato, Marcus

Building a Stationary Stochastic Process From a FiniteDimensional Marginal
If $\mathfrak{A}$ is a finite alphabet, $\sU \subset\mathbb{Z}^D$, and
$\mu_\sU$ is a probability measure on $\mathfrak{A}^\sU$ that ``looks like''
the marginal projection of a stationary stochastic process on
$\mathfrak{A}^{\mathbb{Z}^D}$, then can we ``extend''
$\mu_\sU$ to such a process? Under what conditions can we make this
extension ergodic, (quasi)periodic, or (weakly) mixing? After surveying
classical work on this problem when $D=1$, we provide some sufficient
conditions and some necessary conditions for $\mu_\sU$ to be extendible
for $D>1$, and show that, in general, the problem is not formally decidable.
Categories:37A50, 60G10, 37B10 
