location:  Publications → journals
Search results

Search: MSC category 37 ( Dynamical systems and ergodic theory )

 Expand all        Collapse all Results 1 - 25 of 43

1. CJM Online first

Cahn, Jordan; Jones, Rafe; Spear, Jacob
 Powers in orbits of rational functions: cases of an arithmetic dynamical Mordell-Lang conjecture Let $K$ be a finitely generated field of characteristic zero. We study, for fixed $m \geq 2$, the rational functions $\phi$ defined over $K$ that have a $K$-orbit containing infinitely many distinct $m$th powers. For $m \geq 5$ we show the only such functions are those of the form $cx^j(\psi(x))^m$ with $\psi \in K(x)$, and for $m \leq 4$ we show the only additional cases are certain LattÃ¨s maps and four families of rational functions whose special properties appear not to have been studied before. With additional analysis, we show that the index set $\{n \geq 0 : \phi^{n}(a) \in \lambda(\mathbb{P}^1(K))\}$ is a union of finitely many arithmetic progressions, where $\phi^{n}$ denotes the $n$th iterate of $\phi$ and $\lambda \in K(x)$ is any map MÃ¶bius-conjugate over $K$ to $x^m$. When the index set is infinite, we give bounds on the number and moduli of the arithmetic progressions involved. These results are similar in flavor to the dynamical Mordell-Lang conjecture, and motivate a new conjecture on the intersection of an orbit with the value set of a morphism. A key ingredient in our proofs is a study of the curves $y^m = \phi^{n}(x)$. We describe all $\phi$ for which these curves have an irreducible component of genus at most 1, and show that such $\phi$ must have two distinct iterates that are equal in $K(x)^*/K(x)^{*m}$. Keywords:arithmetic dynamics, iteration of rational functions, special orbits of rational function, genus of variables-separated curve, LattÃ¨s mapCategories:37P05, 11G05, 37P15

2. CJM Online first

Matsumoto, Kengo
 Asymptotic continuous orbit equivalence of Smale spaces and Ruelle algebras In the first part of the paper, we introduce notions of asymptotic continuous orbit equivalence and asymptotic conjugacy in Smale spaces and characterize them in terms of their asymptotic Ruelle algebras with their dual actions. In the second part, we introduce a groupoid $C^*$-algebra which is an extended version of the asymptotic Ruelle algebra from a Smale space and study the extended Ruelle algebras from the view points of Cuntz-Krieger algebras. As a result, the asymptotic Ruelle algebra is realized as a fixed point algebra of the extended Ruelle algebra under certain circle action. Keywords:hyperbolic dynamics, Smale space, Ruelle algebra, groupoid, zeta function, continuous orbit equivalence, shifts of finite type, Cuntz-Krieger algebraCategories:37D20, 46L35

3. CJM Online first

Handelman, David
 Nearly approximate transitivity (AT) for circulant matrices By previous work of Giordano and the author, ergodic actions of $\mathbf Z$ (and other discrete groups) are completely classified measure-theoretically by their dimension space, a construction analogous to the dimension group used in C*-algebras and topological dynamics. Here we investigate how far from AT (approximately transitive) can actions be which derive from circulant (and related) matrices. It turns out not very: although non-AT actions can arise from this method of construction, under very modest additional conditions, ATness arises; in addition, if we drop the positivity requirement in the isomorphism of dimension spaces, then all these ergodic actions satisfy an analogue of AT. Many examples are provided. Keywords:approximately transitive, ergodic transformation, circulant matrix, hemicirculant matrix, dimension space, matrix-valued random walkCategories:37A05, 06F25, 28D05, 46B40, 60G50

4. CJM 2017 (vol 70 pp. 294)

Eilers, Søren; Restorff, Gunnar; Ruiz, Efren; Sørensen, Adam P. W.
 Geometric classification of graph C*-algebras over finite graphs We address the classification problem for graph $C^*$-algebras of finite graphs (finitely many edges and vertices), containing the class of Cuntz-Krieger algebras as a prominent special case. Contrasting earlier work, we do not assume that the graphs satisfy the standard condition (K), so that the graph $C^*$-algebras may come with uncountably many ideals. We find that in this generality, stable isomorphism of graph $C^*$-algebras does not coincide with the geometric notion of Cuntz move equivalence. However, adding a modest condition on the graphs, the two notions are proved to be mutually equivalent and equivalent to the $C^*$-algebras having isomorphic $K$-theories. This proves in turn that under this condition, the graph $C^*$-algebras are in fact classifiable by $K$-theory, providing in particular complete classification when the $C^*$-algebras in question are either of real rank zero or type I/postliminal. The key ingredient in obtaining these results is a characterization of Cuntz move equivalence using the adjacency matrices of the graphs. Our results are applied to discuss the classification problem for the quantum lens spaces defined by Hong and SzymaÅski, and to complete the classification of graph $C^*$-algebras associated to all simple graphs with four vertices or less. Keywords:graph $C^*$-algebra, geometric classification, $K$-theory, flow equivalenceCategories:46L35, 46L80, 46L55, 37B10

5. CJM 2017 (vol 70 pp. 538)

Ciesielski, Krzysztof Chris; Jasinski, Jakub
 Fixed Point Theorems for Maps with Local and Pointwise Contraction Properties The paper constitutes a comprehensive study of ten classes of self-maps on metric spaces $\langle X,d\rangle$ with the local and pointwise (a.k.a. local radial) contraction properties. Each of those classes appeared previously in the literature in the context of fixed point theorems. We begin with presenting an overview of these fixed point results, including concise self contained sketches of their proofs. Then, we proceed with a discussion of the relations among the ten classes of self-maps with domains $\langle X,d\rangle$ having various topological properties which often appear in the theory of fixed point theorems: completeness, compactness, (path) connectedness, rectifiable path connectedness, and $d$-convexity. The bulk of the results presented in this part consists of examples of maps that show non-reversibility of the previously established inclusions between theses classes. Among these examples, the most striking is a differentiable auto-homeomorphism $f$ of a compact perfect subset $X$ of $\mathbb R$ with $f'\equiv 0$, which constitutes also a minimal dynamical system. We finish with discussing a few remaining open problems on weather the maps with specific pointwise contraction properties must have the fixed points. Keywords:fixed point, periodic point, contractive map, locally contractive map, pointwise contractive map, radially contractive map, rectifiably path connected space, d-convex, geodesic, remetrization contraction mapping principleCategories:54H25, 37C25

6. CJM 2017 (vol 70 pp. 218)

Speissegger, Patrick
 Quasianalytic Ilyashenko algebras I construct a quasianalytic field $\mathcal{F}$ of germs at $+\infty$ of real functions with logarithmic generalized power series as asymptotic expansions, such that $\mathcal{F}$ is closed under differentiation and $\log$-composition; in particular, $\mathcal{F}$ is a Hardy field. Moreover, the field $\mathcal{F} \circ (-\log)$ of germs at $0^+$ contains all transition maps of hyperbolic saddles of planar real analytic vector fields. Keywords:generalized series expansion, quasianalyticity, transition mapCategories:41A60, 30E15, 37D99, 03C99

7. CJM 2016 (vol 69 pp. 532)

Ganguly, Arijit; Ghosh, Anish
 Dirichlet's Theorem in Function Fields We study metric Diophantine approximation for function fields specifically the problem of improving Dirichlet's theorem in Diophantine approximation. Keywords:Dirichlet's theorem, Diophantine approximation, positive characteristicCategories:11J83, 11K60, 37D40, 37A17, 22E40

8. CJM 2016 (vol 68 pp. 625)

Ingram, Patrick
 Rigidity and Height Bounds for Certain Post-critically Finite Endomorphisms of $\mathbb P^N$ The morphism $f:\mathbb{P}^N\to\mathbb{P}^N$ is called post-critically finite (PCF) if the forward image of the critical locus, under iteration of $f$, has algebraic support. In the case $N=1$, a result of Thurston implies that there are no algebraic families of PCF morphisms, other than a well-understood exceptional class known as the flexible LattÃ¨s maps. A related arithmetic result states that the set of PCF morphisms corresponds to a set of bounded height in the moduli space of univariate rational functions. We prove corresponding results for a certain subclass of the regular polynomial endomorphisms of $\mathbb{P}^N$, for any $N$. Keywords:post-critically finite, arithmetic dynamics, heightsCategories:37P15, 32H50, 37P30

9. CJM 2015 (vol 67 pp. 1247)

Barros, Carlos Braga; Rocha, Victor; Souza, Josiney
 Lyapunov Stability and Attraction Under Equivariant Maps Let $M$ and $N$ be admissible Hausdorff topological spaces endowed with admissible families of open coverings. Assume that $\mathcal{S}$ is a semigroup acting on both $M$ and $N$. In this paper we study the behavior of limit sets, prolongations, prolongational limit sets, attracting sets, attractors and Lyapunov stable sets (all concepts defined for the action of the semigroup $\mathcal{S}$) under equivariant maps and semiconjugations from $M$ to $N$. Keywords:Lyapunov stability, semigroup actions, generalized flows, equivariant maps, admissible topological spacesCategories:37B25, 37C75, 34C27, 34D05

10. CJM 2015 (vol 67 pp. 1144)

Nystedt, Patrik; Öinert, Johan
 Outer Partial Actions and Partial Skew Group Rings We extend the classicial notion of an outer action $\alpha$ of a group $G$ on a unital ring $A$ to the case when $\alpha$ is a partial action on ideals, all of which have local units. We show that if $\alpha$ is an outer partial action of an abelian group $G$, then its associated partial skew group ring $A \star_\alpha G$ is simple if and only if $A$ is $G$-simple. This result is applied to partial skew group rings associated with two different types of partial dynamical systems. Keywords:outer action, partial action, minimality, topological dynamics, partial skew group ring, simplicityCategories:16W50, 37B05, 37B99, 54H15, 54H20

11. CJM 2014 (vol 67 pp. 795)

Di Nasso, Mauro; Goldbring, Isaac; Jin, Renling; Leth, Steven; Lupini, Martino; Mahlburg, Karl
 On a Sumset Conjecture of ErdÅs ErdÅs conjectured that for any set $A\subseteq \mathbb{N}$ with positive lower asymptotic density, there are infinite sets $B,C\subseteq \mathbb{N}$ such that $B+C\subseteq A$. We verify ErdÅs' conjecture in the case that $A$ has Banach density exceeding $\frac{1}{2}$. As a consequence, we prove that, for $A\subseteq \mathbb{N}$ with positive Banach density (a much weaker assumption than positive lower density), we can find infinite $B,C\subseteq \mathbb{N}$ such that $B+C$ is contained in the union of $A$ and a translate of $A$. Both of the aforementioned results are generalized to arbitrary countable amenable groups. We also provide a positive solution to ErdÅs' conjecture for subsets of the natural numbers that are pseudorandom. Keywords:sumsets of integers, asymptotic density, amenable groups, nonstandard analysisCategories:11B05, 11B13, 11P70, 28D15, 37A45

12. CJM 2014 (vol 67 pp. 90)

Bousch, Thierry
 Une propriÃ©tÃ© de domination convexe pour les orbites sturmiennes Let ${\bf x}=(x_0,x_1,\ldots)$ be a $N$-periodic sequence of integers ($N\ge1$), and ${\bf s}$ a sturmian sequence with the same barycenter (and also $N$-periodic, consequently). It is shown that, for affine functions $\alpha:\mathbb R^\mathbb N_{(N)}\to\mathbb R$ which are increasing relatively to some order $\le_2$ on $\mathbb R^\mathbb N_{(N)}$ (the space of all $N$-periodic sequences), the average of $|\alpha|$ on the orbit of ${\bf x}$ is greater than its average on the orbit of ${\bf s}$. Keywords:suite sturmienne, domination convexe, optimisation ergodiqueCategories:37D35, 49N20, 90C27

13. CJM 2014 (vol 67 pp. 330)

Bernardes, Nilson C.; Vermersch, Rômulo M.
 Hyperspace Dynamics of Generic Maps of the Cantor Space We study the hyperspace dynamics induced from generic continuous maps and from generic homeomorphisms of the Cantor space, with emphasis on the notions of Li-Yorke chaos, distributional chaos, topological entropy, chain continuity, shadowing and recurrence. Keywords:cantor space, continuous maps, homeomorphisms, hyperspace, dynamicsCategories:37B99, 54H20, 54E52

14. CJM 2013 (vol 66 pp. 57)

Bezuglyi, S.; Kwiatkowski, J.; Yassawi, R.
 Perfect Orderings on Finite Rank Bratteli Diagrams Given a Bratteli diagram $B$, we study the set $\mathcal O_B$ of all possible orderings on $B$ and its subset $\mathcal P_B$ consisting of perfect orderings that produce Bratteli-Vershik topological dynamical systems (Vershik maps). We give necessary and sufficient conditions for the ordering $\omega$ to be perfect. On the other hand, a wide class of non-simple Bratteli diagrams that do not admit Vershik maps is explicitly described. In the case of finite rank Bratteli diagrams, we show that the existence of perfect orderings with a prescribed number of extreme paths constrains significantly the values of the entries of the incidence matrices and the structure of the diagram $B$. Our proofs are based on the new notions of skeletons and associated graphs, defined and studied in the paper. For a Bratteli diagram $B$ of rank $k$, we endow the set $\mathcal O_B$ with product measure $\mu$ and prove that there is some $1 \leq j\leq k$ such that $\mu$-almost all orderings on $B$ have $j$ maximal and $j$ minimal paths. If $j$ is strictly greater than the number of minimal components that $B$ has, then $\mu$-almost all orderings are imperfect. Keywords:Bratteli diagrams, Vershik mapsCategories:37B10, 37A20

15. CJM 2013 (vol 65 pp. 1287)

Reihani, Kamran
 $K$-theory of Furstenberg Transformation Group $C^*$-algebras The paper studies the $K$-theoretic invariants of the crossed product $C^{*}$-algebras associated with an important family of homeomorphisms of the tori $\mathbb{T}^{n}$ called Furstenberg transformations. Using the Pimsner-Voiculescu theorem, we prove that given $n$, the $K$-groups of those crossed products, whose corresponding $n\times n$ integer matrices are unipotent of maximal degree, always have the same rank $a_{n}$. We show using the theory developed here that a claim made in the literature about the torsion subgroups of these $K$-groups is false. Using the representation theory of the simple Lie algebra $\frak{sl}(2,\mathbb{C})$, we show that, remarkably, $a_{n}$ has a combinatorial significance. For example, every $a_{2n+1}$ is just the number of ways that $0$ can be represented as a sum of integers between $-n$ and $n$ (with no repetitions). By adapting an argument of van Lint (in which he answered a question of ErdÅs), a simple, explicit formula for the asymptotic behavior of the sequence $\{a_{n}\}$ is given. Finally, we describe the order structure of the $K_{0}$-groups of an important class of Furstenberg crossed products, obtaining their complete Elliott invariant using classification results of H. Lin and N. C. Phillips. Keywords:$K$-theory, transformation group $C^*$-algebra, Furstenberg transformation, Anzai transformation, minimal homeomorphism, positive cone, minimal homeomorphismCategories:19K14, 19K99, 46L35, 46L80, , 05A15, 05A16, 05A17, 15A36, 17B10, 17B20, 37B05, 54H20

16. CJM 2013 (vol 67 pp. 450)

Santoprete, Manuele; Scheurle, Jürgen; Walcher, Sebastian
 Motion in a Symmetric Potential on the Hyperbolic Plane We study the motion of a particle in the hyperbolic plane (embedded in Minkowski space), under the action of a potential that depends only on one variable. This problem is the analogous to the spherical pendulum in a unidirectional force field. However, for the discussion of the hyperbolic plane one has to distinguish three inequivalent cases, depending on the direction of the force field. Symmetry reduction, with respect to groups that are not necessarily compact or even reductive, is carried out by way of Poisson varieties and Hilbert maps. For each case the dynamics is discussed, with special attention to linear potentials. Keywords:Hamiltonian systems with symmetry, symmetries, non-compact symmetry groups, singular reductionCategories:37J15, 70H33, 70F99, 37C80, 34C14, , 20G20

17. CJM 2013 (vol 65 pp. 1384)

Wright, Paul
 Estimates of Hausdorff Dimension for Non-wandering Sets of Higher Dimensional Open Billiards This article concerns a class of open billiards consisting of a finite number of strictly convex, non-eclipsing obstacles $K$. The non-wandering set $M_0$ of the billiard ball map is a topological Cantor set and its Hausdorff dimension has been previously estimated for billiards in $\mathbb{R}^2$, using well-known techniques. We extend these estimates to billiards in $\mathbb{R}^n$, and make various refinements to the estimates. These refinements also allow improvements to other results. We also show that in many cases, the non-wandering set is confined to a particular subset of $\mathbb{R}^n$ formed by the convex hull of points determined by period 2 orbits. This allows more accurate bounds on the constants used in estimating Hausdorff dimension. Keywords:dynamical systems, billiards, dimension, HausdorffCategories:37D20, 37D40

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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
 Page 1 2 Next
 top of page | contact us | privacy | site map |