1. CJM Online first
 Cahn, Jordan; Jones, Rafe; Spear, Jacob

Powers in orbits of rational functions: cases of an arithmetic dynamical MordellLang 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¶biusconjugate 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
MordellLang 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 variablesseparated curve, LattÃ¨s map Categories: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
CuntzKrieger 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, CuntzKrieger algebra Categories: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
measuretheoretically 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 nonAT 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, matrixvalued random walk Categories: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 CuntzKrieger 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 SzymaÅski,
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 equivalence Categories: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
selfmaps 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 selfmaps 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 nonreversibility of the previously established
inclusions between theses classes.
Among these examples, the most striking is a differentiable autohomeomorphism
$f$ 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, dconvex, geodesic, remetrization contraction mapping principle Categories: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 map Categories: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 characteristic Categories:11J83, 11K60, 37D40, 37A17, 22E40 

8. CJM 2016 (vol 68 pp. 625)
 Ingram, Patrick

Rigidity and Height Bounds for Certain Postcritically Finite Endomorphisms of $\mathbb P^N$
The morphism $f:\mathbb{P}^N\to\mathbb{P}^N$ is called postcritically 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 wellunderstood 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:postcritically finite, arithmetic dynamics, heights Categories: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 spaces Categories: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, simplicity Categories: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 ErdÅs
ErdÅs 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 ErdÅs' 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 ErdÅs'
conjecture for subsets of the natural numbers that are pseudorandom.
Keywords:sumsets of integers, asymptotic density, amenable groups, nonstandard analysis Categories: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 ergodique Categories: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 LiYorke chaos, distributional chaos, topological entropy,
chain continuity, shadowing and recurrence.
Keywords:cantor space, continuous maps, homeomorphisms, hyperspace, dynamics Categories: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
BratteliVershik 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 nonsimple 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 maps Categories: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 PimsnerVoiculescu 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 ErdÅs), 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 homeomorphism Categories: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, noncompact symmetry groups, singular reduction Categories:37J15, 70H33, 70F99, 37C80, 34C14, , 20G20 

17. CJM 2013 (vol 65 pp. 1384)
 Wright, Paul

Estimates of Hausdorff Dimension for Nonwandering Sets of Higher Dimensional Open Billiards
This article concerns a class of open billiards consisting of a finite
number of strictly convex, noneclipsing obstacles $K$. The
nonwandering 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 wellknown 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
nonwandering 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, Hausdorff Categories:37D20, 37D40 

18. CJM 2012 (vol 65 pp. 553)
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, secondcountable 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 finiterange 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 systems Categories: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 dynamics Category: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 systems Category: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 entropymaximizing, 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 space Categories: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 surface Categories:37D50, 49Q10 

24. CJM 2011 (vol 63 pp. 1201)
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, dynamics Categories:14J28, , , , 14J50, 11D41, 11D72, 11H56, 11G10, 37F35, 37D05 
