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1. CJM 2012 (vol 65 pp. 553)

Godinho, Leonor; Sousa-Dias, M. E.
 Addendum and Erratum to "The Fundamental Group of $S^1$-manifolds" This paper provides an addendum and erratum to L. Godinho and M. E. Sousa-Dias, "The Fundamental Group of $S^1$-manifolds". Canad. J. Math. 62(2010), no. 5, 1082--1098. Keywords:symplectic reduction; fundamental groupCategories:53D19, 37J10, 55Q05

2. CJM 2012 (vol 65 pp. 349)

Müller, Peter; Richard, Christoph
 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 systemsCategories:37B50, 37A30

3. CJM 2012 (vol 64 pp. 318)

Ingram, Patrick
 Cubic Polynomials with Periodic Cycles of a Specified Multiplier We consider cubic polynomials $f(z)=z^3+az+b$ defined over $\mathbb{C}(\lambda)$, with a marked point of period $N$ and multiplier $\lambda$. In the case $N=1$, there are infinitely many such objects, and in the case $N\geq 3$, only finitely many (subject to a mild assumption). The case $N=2$ has particularly rich structure, and we are able to describe all such cubic polynomials defined over the field $\bigcup_{n\geq 1}\mathbb{C}(\lambda^{1/n})$. Keywords:cubic polynomials, periodic points, holomorphic dynamicsCategory:37P35

4. CJM 2011 (vol 65 pp. 149)

Kellendonk, Johannes; Lenz, Daniel
 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 systemsCategory:37B50

5. CJM 2011 (vol 64 pp. 1341)

Killough, D. B.; Putnam, I. F.
 Bowen Measure From Heteroclinic Points We present a new construction of the entropy-maximizing, invariant probability measure on a Smale space (the Bowen measure). Our construction is based on points that are unstably equivalent to one given point, and stably equivalent to another: heteroclinic points. The spirit of the construction is similar to Bowen's construction from periodic points, though the techniques are very different. We also prove results about the growth rate of certain sets of heteroclinic points, and about the stable and unstable components of the Bowen measure. The approach we take is to prove results through direct computation for the case of a Shift of Finite type, and then use resolving factor maps to extend the results to more general Smale spaces. Keywords:hyperbolic dynamics, Smale spaceCategories:37D20, 37B10

6. CJM 2011 (vol 64 pp. 1058)

Plakhov, Alexander
 Optimal Roughening of Convex Bodies A body moves in a rarefied medium composed of point particles at rest. The particles make elastic reflections when colliding with the body surface, and do not interact with each other. We consider a generalization of Newton's minimal resistance problem: given two bounded convex bodies $C_1$ and $C_2$ such that $C_1 \subset C_2 \subset \mathbb{R}^3$ and $\partial C_1 \cap \partial C_2 = \emptyset$, minimize the resistance in the class of connected bodies $B$ such that $C_1 \subset B \subset C_2$. We prove that the infimum of resistance is zero; that is, there exist "almost perfectly streamlined" bodies. Keywords:billiards, shape optimization, problems of minimal resistance, Newtonian aerodynamics, rough surfaceCategories:37D50, 49Q10

7. CJM 2011 (vol 63 pp. 1201)

Abou Salem, Walid K. ; Sulem, Catherine
 Resonant Tunneling of Fast Solitons through Large Potential Barriers We rigorously study the resonant tunneling of fast solitons through large potential barriers for the nonlinear SchrÃ¶dinger equation in one dimension. Our approach covers the case of general nonlinearities, both local and Hartree (nonlocal). Keywords:nonlinear Schroedinger equations, external potential, solitary waves, long time behavior, resonant tunnelingCategories:37K40, 35Q55, 35Q51

8. CJM 2011 (vol 63 pp. 481)

Baragar, Arthur
 The Ample Cone for a K3 Surface In this paper, we give several pictorial fractal representations of the ample or KÃ¤hler cone for surfaces in a certain class of $K3$ surfaces. The class includes surfaces described by smooth $(2,2,2)$ forms in ${\mathbb P^1\times\mathbb P^1\times \mathbb P^1}$ defined over a sufficiently large number field $K$ that have a line parallel to one of the axes and have Picard number four. We relate the Hausdorff dimension of this fractal to the asymptotic growth of orbits of curves under the action of the surface's group of automorphisms. We experimentally estimate the Hausdorff dimension of the fractal to be $1.296 \pm .010$. Keywords:Fractal, Hausdorff dimension, K3 surface, Kleinian groups, dynamicsCategories:14J28, , , , 14J50, 11D41, 11D72, 11H56, 11G10, 37F35, 37D05

9. CJM 2010 (vol 62 pp. 1082)

Godinho, Leonor; Sousa-Dias, M. E.
 The Fundamental Group of $S^1$-manifolds We address the problem of computing the fundamental group of a symplectic $S^1$-manifold for non-Hamiltonian actions on compact manifolds, and for Hamiltonian actions on non-compact manifolds with a proper moment map. We generalize known results for compact manifolds equipped with a Hamiltonian $S^1$-action. Several examples are presented to illustrate our main results. Categories:53D20, 37J10, 55Q05

10. CJM 2009 (vol 61 pp. 656)

McCutcheon, Randall; Quas, Anthony
 Generalized Polynomials and Mild Mixing An unsettled conjecture of V. Bergelson and I. H\aa land proposes that if $(X,\alg,\mu,T)$ is an invertible weak mixing measure preserving system, where $\mu(X)<\infty$, and if $p_1,p_2,\dots ,p_k$ are generalized polynomials (functions built out of regular polynomials via iterated use of the greatest integer or floor function) having the property that no $p_i$, nor any $p_i-p_j$, $i\neq j$, is constant on a set of positive density, then for any measurable sets $A_0,A_1,\dots ,A_k$, there exists a zero-density set $E\subset \z$ such that $\lim_{\substack{n\to\infty\\ n\not\in E}} \,\mu(A_0\cap T^{p_1(n)}A_1\cap \cdots \cap T^{p_k(n)}A_k)=\prod_{i=0}^k \mu(A_i).$ We formulate and prove a faithful version of this conjecture for mildly mixing systems and partially characterize, in the degree two case, the set of families $\{ p_1,p_2, \dots ,p_k\}$ satisfying the hypotheses of this theorem. Categories:37A25, 28D05

11. CJM 2008 (vol 60 pp. 975)

Boca, Florin P.
 An AF Algebra Associated with the Farey Tessellation We associate with the Farey tessellation of the upper half-plane an AF algebra $\AA$ encoding the cutting sequences'' that define vertical geodesics. The Effros--Shen AF algebras arise as quotients of $\AA$. Using the path algebra model for AF algebras we construct, for each $\tau \in \big(0,\frac{1}{4}\big]$, projections $(E_n)$ in $\AA$ such that $E_n E_{n\pm 1}E_n \leq \tau E_n$. Categories:46L05, 11A55, 11B57, 46L55, 37E05, 82B20

12. CJM 2008 (vol 60 pp. 572)

Hitrik, Michael; Sj{östrand, Johannes
 Non-Selfadjoint Perturbations of Selfadjoint Operators in Two Dimensions IIIa. One Branching Point This is the third in a series of works devoted to spectral asymptotics for non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2, having a periodic classical flow. Assuming that the strength $\epsilon$ of the perturbation is in the range $h^2\ll \epsilon \ll h^{1/2}$ (and may sometimes reach even smaller values), we get an asymptotic description of the eigenvalues in rectangles $[-1/C,1/C]+i\epsilon [F_0-1/C,F_0+1/C]$, $C\gg 1$, when $\epsilon F_0$ is a saddle point value of the flow average of the leading perturbation. Keywords:non-selfadjoint, eigenvalue, periodic flow, branching singularityCategories:31C10, 35P20, 35Q40, 37J35, 37J45, 53D22, 58J40

13. CJM 2008 (vol 60 pp. 658)

Mihailescu, Eugen; Urba\'nski, Mariusz
 Inverse Pressure Estimates and the Independence of Stable Dimension for Non-Invertible Maps We study the case of an Axiom A holomorphic non-degenerate (hence non-invertible) map $f\from\mathbb P^2 \mathbb C \to \mathbb P^2 \mathbb C$, where $\mathbb P^2 \mathbb C$ stands for the complex projective space of dimension 2. Let $\Lambda$ denote a basic set for $f$ of unstable index 1, and $x$ an arbitrary point of $\Lambda$; we denote by $\delta^s(x)$ the Hausdorff dimension of $W^s_r(x) \cap \Lambda$, where $r$ is some fixed positive number and $W^s_r(x)$ is the local stable manifold at $x$ of size $r$; $\delta^s(x)$ is called \emph{the stable dimension at} $x$. Mihailescu and Urba\'nski introduced a notion of inverse topological pressure, denoted by $P^-$, which takes into consideration preimages of points. Manning and McCluskey study the case of hyperbolic diffeomorphisms on real surfaces and give formulas for Hausdorff dimension. Our non-invertible situation is different here since the local unstable manifolds are not uniquely determined by their base point, instead they depend in general on whole prehistories of the base points. Hence our methods are different and are based on using a sequence of inverse pressures for the iterates of $f$, in order to give upper and lower estimates of the stable dimension. We obtain an estimate of the oscillation of the stable dimension on $\Lambda$. When each point $x$ from $\Lambda$ has the same number $d'$ of preimages in $\Lambda$, then we show that $\delta^s(x)$ is independent of $x$; in fact $\delta^s(x)$ is shown to be equal in this case with the unique zero of the map $t \to P(t\phi^s - \log d')$. We also prove the Lipschitz continuity of the stable vector spaces over $\Lambda$; this proof is again different than the one for diffeomorphisms (however, the unstable distribution is not always Lipschitz for conformal non-invertible maps). In the end we include the corresponding results for a real conformal setting. Keywords:Hausdorff dimension, stable manifolds, basic sets, inverse topological pressureCategories:37D20, 37A35, 37F35

14. CJM 2008 (vol 60 pp. 189)

Lin, Huaxin
 Furstenberg Transformations and Approximate Conjugacy Let $\alpha$ and $\beta$ be two Furstenberg transformations on $2$-torus associated with irrational numbers $\theta_1,$ $\theta_2,$ integers $d_1, d_2$ and Lipschitz functions $f_1$ and $f_2$. It is shown that $\alpha$ and $\beta$ are approximately conjugate in a measure theoretical sense if (and only if) $\overline{\theta_1\pm \theta_2}=0$ in $\R/\Z.$ Closely related to the classification of simple amenable \CAs, it is shown that $\af$ and $\bt$ are approximately $K$-conjugate if (and only if) $\overline{\theta_1\pm \theta_2}=0$ in $\R/\Z$ and $|d_1|=|d_2|.$ This is also shown to be equivalent to the condition that the associated crossed product \CAs are isomorphic. Keywords:Furstenberg transformations, approximate conjugacyCategories:37A55, 46L35

15. CJM 2007 (vol 59 pp. 596)

Itzá-Ortiz, Benjamín A.
 Eigenvalues, $K$-theory and Minimal Flows Let $(Y,T)$ be a minimal suspension flow built over a dynamical system $(X,S)$ and with (strictly positive, continuous) ceiling function $f\colon X\to\R$. We show that the eigenvalues of $(Y,T)$ are contained in the range of a trace on the $K_0$-group of $(X,S)$. Moreover, a trace gives an order isomorphism of a subgroup of $K_0(\cprod{C(X)}{S})$ with the group of eigenvalues of $(Y,T)$. Using this result, we relate the values of $t$ for which the time-$t$ map on the minimal suspension flow is minimal with the $K$-theory of the base of this suspension. Categories:37A55, 37B05

16. 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 dynamicsCategory:37C30

17. 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

18. 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

19. CJM 2004 (vol 56 pp. 553)

 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

20. 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. Category:37A05

21. 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

22. 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

23. 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, MÃ¶bius inversionCategories:92D10, 34L30, 37N30, 06A07, 60J25

24. 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

25. 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 manifoldsCategories:37J15, 58A40, 58D19, 70H33
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