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26. CJM 2007 (vol 59 pp. 311)

Christianson, Hans
Growth and Zeros of the Zeta Function for Hyperbolic Rational Maps
This paper describes new results on the growth and zeros of the Ruelle zeta function for the Julia set of a hyperbolic rational map. It is shown that the zeta function is bounded by $\exp(C_K |s|^{\delta})$ in strips $|\Real s| \leq K$, where $\delta$ is the dimension of the Julia set. This leads to bounds on the number of zeros in strips (interpreted as the Pollicott--Ruelle resonances of this dynamical system). An upper bound on the number of zeros in polynomial regions $\{|\Real s | \leq |\Imag s|^\alpha\}$ is given, followed by weaker lower bound estimates in strips $\{\Real s > -C, |\Imag s|\leq r\}$, and logarithmic neighbourhoods $\{ |\Real s | \leq \rho \log |\Imag s| \}$. Recent numerical work of Strain--Zworski suggests the upper bounds in strips are optimal.

Keywords:zeta function, transfer operator, complex dynamics

27. 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 measure-preserving transformation of a measure space or a covering map of a compact space, are special cases of the endomorphism crossed-product 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

28. 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

29. 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 Tolman-Weitsman 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 Hausmann-Knutson are special cases of our result.

Categories:53D20, 53D30, 37J10, 37J15, 53D05

30. CJM 2004 (vol 56 pp. 449)

Demeter, Ciprian
The Best Constants Associated with Some Weak Maximal Inequalities in Ergodic Theory
We introduce a new device of measuring the degree of the failure of convergence in the ergodic theorem along subsequences of integers. Relations with other types of bad behavior in ergodic theory and applications to weighted averages are also discussed.


31. CJM 2004 (vol 56 pp. 115)

Kenny, Robert
Estimates of Hausdorff Dimension for the Non-Wandering 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 well-known 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

32. CJM 2003 (vol 55 pp. 247)

Cushman, Richard; Śniatycki, Jędrzej
Differential Structure of Orbit Spaces: Erratum
This note signals an error in the above paper by giving a counter-example.

Categories:37J15, 58A40, 58D19, 70H33

33. 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 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, Möbius inversion
Categories:92D10, 34L30, 37N30, 06A07, 60J25

34. 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

35. 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 Stefan-Sussmann 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

36. CJM 2001 (vol 53 pp. 382)

Pivato, Marcus
Building a Stationary Stochastic Process From a Finite-Dimensional 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
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