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1. CMB Online first

Ingram, Patrick
 $p$-adic uniformization and the action of Galois on certain affine correspondences Given two monic polynomials $f$ and $g$ with coefficients in a number field $K$, and some $\alpha\in K$, we examine the action of the absolute Galois group $\operatorname{Gal}(\overline{K}/K)$ on the directed graph of iterated preimages of $\alpha$ under the correspondence $g(y)=f(x)$, assuming that $\deg(f)\gt \deg(g)$ and that $\gcd(\deg(f), \deg(g))=1$. If a prime of $K$ exists at which $f$ and $g$ have integral coefficients, and at which $\alpha$ is not integral, we show that this directed graph of preimages consists of finitely many $\operatorname{Gal}(\overline{K}/K)$-orbits. We obtain this result by establishing a $p$-adic uniformization of such correspondences, tenuously related to BÃ¶ttcher's uniformization of polynomial dynamical systems over $\mathbb{CC}$, although the construction of a BÃ¶ttcher coordinate for complex holomorphic correspondences remains unresolved. Keywords:arithmetic dynamicsCategories:37P20, 11S20

2. CMB Online first

Aujogue, Jean-Baptiste
 A short proof of the characterization of model sets by almost automorphy The aim of this note is to provide a conceptually simple demonstration of the fact that repetitive model sets are characterized as the repetitive Meyer sets with an almost automorphic associated dynamical system. Keywords:Meyer set, model set, almost automorphyCategories:37B50, 37B05

3. CMB 2017 (vol 61 pp. 149)

Llibre, Jaume; Valls, Claudia
 Global phase portraits for the Abel quadratic polynomial differential equations of second kind with $Z_2$-symmetries We provide normal forms and the global phase portraits on the PoincarÃ© disk for all Abel quadratic polynomial differential equations of the second kind with $\mathbb Z_2$-symmetries. Keywords:Abel polynomial differential system of the second kind, vector field, phase portraitCategories:37J35, 37K10

4. CMB 2016 (vol 60 pp. 411)

Stoyanov, Luchezar
 On Gibbs Measures and Spectra of Ruelle Transfer Operators We prove a comprehensive version of the Ruelle-Perron-Frobenius Theorem with explicit estimates of the spectral radius of the Ruelle transfer operator and various other quantities related to spectral properties of this operator. The novelty here is that the HÃ¶lder constant of the function generating the operator appears only polynomially, not exponentially as in previous known estimates. Keywords:subshift of finite type, Ruelle transfer operator, Gibbs measureCategories:37A05, 37B10

5. CMB 2015 (vol 59 pp. 95)

Gonçalves, Daniel; Li, Hui; Royer, Danilo
 Faithful Representations of Graph Algebras via Branching Systems We continue to investigate branching systems of directed graphs and their connections with graph algebras. We give a sufficient condition under which the representation induced from a branching system of a directed graph is faithful and construct a large class of branching systems that satisfy this condition. We finish the paper by providing a proof of the converse of the Cuntz-Krieger uniqueness theorem for graph algebras by means of branching systems. Keywords:C*-algebra, graph algebra, Leavitt path algebra, branching system, representationCategories:46L05, 37A55

6. CMB 2014 (vol 57 pp. 511)

Gonçalves, Daniel
 Simplicity of Partial Skew Group Rings of Abelian Groups Let $A$ be a ring with local units, $E$ a set of local units for $A$, $G$ an abelian group and $\alpha$ a partial action of $G$ by ideals of $A$ that contain local units. We show that $A\star_{\alpha} G$ is simple if and only if $A$ is $G$-simple and the center of the corner $e\delta_0 (A\star_{\alpha} G) e \delta_0$ is a field for all $e\in E$. We apply the result to characterize simplicity of partial skew group rings in two cases, namely for partial skew group rings arising from partial actions by clopen subsets of a compact set and partial actions on the set level. Keywords:partial skew group rings, simple rings, partial actions, abelian groupsCategories:16S35, 37B05

7. CMB 2012 (vol 57 pp. 240)

Bernardes, Nilson C.
 Addendum to Limit Sets of Typical Homeomorphisms'' Given an integer $n \geq 3$, a metrizable compact topological $n$-manifold $X$ with boundary, and a finite positive Borel measure $\mu$ on $X$, we prove that for the typical homeomorphism $f : X \to X$, it is true that for $\mu$-almost every point $x$ in $X$ the restriction of $f$ (respectively of $f^{-1}$) to the omega limit set $\omega(f,x)$ (respectively to the alpha limit set $\alpha(f,x)$) is topologically conjugate to the universal odometer. Keywords:topological manifolds, homeomorphisms, measures, Baire category, limit setsCategories:37B20, 54H20, 28C15, 54C35, 54E52

8. CMB 2012 (vol 56 pp. 477)

 Hypercyclic Abelian Groups of Affine Maps on $\mathbb{C}^{n}$ We give a characterization of hypercyclic abelian group $\mathcal{G}$ of affine maps on $\mathbb{C}^{n}$. If $\mathcal{G}$ is finitely generated, this characterization is explicit. We prove in particular that no abelian group generated by $n$ affine maps on $\mathbb{C}^{n}$ has a dense orbit. Keywords:affine, hypercyclic, dense, orbit, affine group, abelianCategories:37C85, 47A16

9. CMB 2012 (vol 56 pp. 709)

Bartošová, Dana
 Universal Minimal Flows of Groups of Automorphisms of Uncountable Structures It is a well-known fact, that the greatest ambit for a topological group $G$ is the Samuel compactification of $G$ with respect to the right uniformity on $G.$ We apply the original description by Samuel from 1948 to give a simple computation of the universal minimal flow for groups of automorphisms of uncountable structures using FraÃ¯ssÃ© theory and Ramsey theory. This work generalizes some of the known results about countable structures. Keywords:universal minimal flows, ultrafilter flows, Ramsey theoryCategories:37B05, 03E02, 05D10, 22F50, 54H20

10. CMB 2011 (vol 56 pp. 136)

 On Constructing Ergodic Hyperfinite Equivalence Relations of Non-Product Type Product type equivalence relations are hyperfinite measured equivalence relations, which, up to orbit equivalence, are generated by product type odometer actions. We give a concrete example of a hyperfinite equivalence relation of non-product type, which is the tail equivalence on a Bratteli diagram. In order to show that the equivalence relation constructed is not of product type we will use a criterion called property A. This property, introduced by Krieger for non-singular transformations, is defined directly for hyperfinite equivalence relations in this paper. Keywords:property A, hyperfinite equivalence relation, non-product typeCategories:37A20, 37A35, 46L10

11. CMB 2011 (vol 55 pp. 858)

von Renesse, Max-K.
 An Optimal Transport View of SchrÃ¶dinger's Equation We show that the SchrÃ¶dinger equation is a lift of Newton's third law of motion $\nabla^\mathcal W_{\dot \mu} \dot \mu = -\nabla^\mathcal W F(\mu)$ on the space of probability measures, where derivatives are taken with respect to the Wasserstein Riemannian metric. Here the potential $\mu \to F(\mu)$ is the sum of the total classical potential energy $\langle V,\mu\rangle$ of the extended system and its Fisher information $\frac {\hbar^2} 8 \int |\nabla \ln \mu |^2 \,d\mu$. The precise relation is established via a well-known (Madelung) transform which is shown to be a symplectic submersion of the standard symplectic structure of complex valued functions into the canonical symplectic space over the Wasserstein space. All computations are conducted in the framework of Otto's formal Riemannian calculus for optimal transportation of probability measures. Keywords:SchrÃ¶dinger equation, optimal transport, Newton's law, symplectic submersionCategories:81C25, 82C70, 37K05

12. CMB 2011 (vol 55 pp. 708)

Demeter, Ciprian
 Improved Range in the Return Times Theorem We prove that the Return Times Theorem holds true for pairs of $L^p-L^q$ functions, whenever $\frac{1}{p}+\frac{1}{q}<\frac{3}{2}$. Keywords:Return Times Theorem, maximal multiplier, maximal inequalityCategories:42B25, 37A45

13. CMB 2011 (vol 55 pp. 297)

Glasner, Eli
 The Group $\operatorname{Aut}(\mu)$ is Roelcke Precompact Following a similar result of Uspenskij on the unitary group of a separable Hilbert space, we show that, with respect to the lower (or Roelcke) uniform structure, the Polish group $G= \operatorname{Aut}(\mu)$ of automorphisms of an atomless standard Borel probability space $(X,\mu)$ is precompact. We identify the corresponding compactification as the space of Markov operators on $L_2(\mu)$ and deduce that the algebra of right and left uniformly continuous functions, the algebra of weakly almost periodic functions, and the algebra of Hilbert functions on $G$, i.e., functions on $G$ arising from unitary representations, all coincide. Again following Uspenskij, we also conclude that $G$ is totally minimal. Keywords:Roelcke precompact, unitary group, measure preserving transformations, Markov operators, weakly almost periodic functionsCategories:54H11, 22A05, 37B05, 54H20

14. CMB 2011 (vol 55 pp. 225)

Bernardes, Nilson C.
 Limit Sets of Typical Homeomorphisms Given an integer $n \geq 3$, a metrizable compact topological $n$-manifold $X$ with boundary, and a finite positive Borel measure $\mu$ on $X$, we prove that for the typical homeomorphism $f \colon X \to X$, it is true that for $\mu$-almost every point $x$ in $X$ the limit set $\omega(f,x)$ is a Cantor set of Hausdorff dimension zero, each point of $\omega(f,x)$ has a dense orbit in $\omega(f,x)$, $f$ is non-sensitive at each point of $\omega(f,x)$, and the function $a \to \omega(f,a)$ is continuous at $x$. Keywords:topological manifolds, homeomorphisms, measures, Baire category, limit setsCategories:37B20, 54H20, 28C15, 54C35, 54E52

15. CMB 2011 (vol 54 pp. 676)

Hammerlindl, Andy
 Quasi-isometry and Plaque Expansiveness We show that a partially hyperbolic diffeomorphism is plaque expansive (a form of structural stability for its center foliation) if the strong stable and unstable foliations are quasi-isometric in the universal cover. In particular, all partially hyperbolic diffeomorphisms on the 3-torus are plaque expansive. Category:37D30

16. CMB 2011 (vol 54 pp. 311)

Marzougui, Habib
 Some Remarks Concerning the Topological Characterization of Limit Sets for Surface Flows We give some extension to theorems of JimÃ©nez LÃ³pez and Soler LÃ³pez concerning the topological characterization for limit sets of continuous flows on closed orientable surfaces. Keywords:flows on surfaces, orbits, class of an orbit, singularities, minimal set, limit set, regular cylinder Categories:37B20, 37E35

17. CMB 2009 (vol 53 pp. 295)

Guo, Boling; Huo, Zhaohui
 The Global Attractor of a Damped, Forced Hirota Equation in $H^1$ The existence of the global attractor of a damped forced Hirota equation in the phase space $H^1(\mathbb R)$ is proved. The main idea is to establish the so-called asymptotic compactness property of the solution operator by energy equation approach. Keywords:global attractor, Fourier restriction norm, damping system, asymptotic compactnessCategories:35Q53, 35B40, 35B41, 37L30

18. CMB 2008 (vol 51 pp. 545)

Ionescu, Marius; Watatani, Yasuo
 $C^{\ast}$-Algebras Associated with Mauldin--Williams Graphs A Mauldin--Williams graph $\mathcal{M}$ is a generalization of an iterated function system by a directed graph. Its invariant set $K$ plays the role of the self-similar set. We associate a $C^{*}$-algebra $\mathcal{O}_{\mathcal{M}}(K)$ with a Mauldin--Williams graph $\mathcal{M}$ and the invariant set $K$, laying emphasis on the singular points. We assume that the underlying graph $G$ has no sinks and no sources. If $\mathcal{M}$ satisfies the open set condition in $K$, and $G$ is irreducible and is not a cyclic permutation, then the associated $C^{*}$-algebra $\mathcal{O}_{\mathcal{M}}(K)$ is simple and purely infinite. We calculate the $K$-groups for some examples including the inflation rule of the Penrose tilings. Categories:46L35, 46L08, 46L80, 37B10

19. CMB 2007 (vol 50 pp. 418)

Matui, Hiroki
 A Short Proof of Affability for Certain Cantor Minimal $\Z^2$-Systems We will show that any extension of a product of two Cantor minimal $\Z$-systems is affable in the sense of Giordano, Putnam and Skau. Category:37B05

20. CMB 2006 (vol 49 pp. 203)

Çömez, Doğan
 The Ergodic Hilbert Transform for Admissible Processes It is shown that the ergodic Hilbert transform exists for a class of bounded symmetric admissible processes relative to invertible measure preserving transformations. This generalizes the well-known result on the existence of the ergodic Hilbert transform. Keywords:Hilbert transform, admissible processesCategories:28D05, 37A99

21. CMB 2005 (vol 48 pp. 302)

Yokonuma, Takeo
 Discrete Sets and Associated Dynamical\\ Systems in a Non-Commutative Setting We define a uniform structure on the set of discrete sets of a locally compact topological space on which a locally compact topological group acts continuously. Then we investigate the completeness of these uniform spaces and study these spaces by means of topological dynamical systems. Categories:52C23, 37B50

22. CMB 2005 (vol 48 pp. 3)

Burq, N.
 Quantum Ergodicity of Boundary Values of Eigenfunctions: A Control Theory Approach Consider $M$, a bounded domain in ${\mathbb R}^d$, which is a Riemanian manifold with piecewise smooth boundary and suppose that the billiard associated to the geodesic flow reflecting on the boundary according to the laws of geometric optics is ergodic. We prove that the boundary value of the eigenfunctions of the Laplace operator with reasonable boundary conditions are asymptotically equidistributed in the boundary, extending previous results by G\'erard and Leichtnam as well as Hassell and Zelditch, obtained under the additional assumption of the convexity of~$M$. Categories:35Q55, 35BXX, 37K05, 37L50, 81Q20

23. CMB 2004 (vol 47 pp. 553)

Kerr, David
 A Geometric Approach to Voiculescu-Brown Entropy A basic problem in dynamics is to identify systems with positive entropy, i.e., systems which are chaotic.'' While there is a vast collection of results addressing this issue in topological dynamics, the phenomenon of positive entropy remains by and large a mystery within the broader noncommutative domain of $C^*$-algebraic dynamics. To shed some light on the noncommutative situation we propose a geometric perspective inspired by work of Glasner and Weiss on topological entropy. This is a written version of the author's talk at the Winter 2002 Meeting of the Canadian Mathematical Society in Ottawa, Ontario. Categories:46L55, 37B40

24. 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 recurrenceCategories:57M50, 37B20

25. 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
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