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26. CMB 2004 (vol 47 pp. 332)

Charette, Virginie; Goldman, William M.; Jones, Catherine A.
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

27. CMB 2004 (vol 47 pp. 168)

Baake, Michael; Sing, Bernd
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

28. CMB 2003 (vol 46 pp. 277)

Rochon, Frédéric
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

29. CMB 2002 (vol 45 pp. 697)

Sirvent, V. F.; Solomyak, B.
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

30. CMB 2002 (vol 45 pp. 123)

Moody, Robert V.
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

31. CMB 2001 (vol 44 pp. 292)

McKay, Angela
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.

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