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Search: All articles in the CJM digital archive with keyword dynamics

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1. CJM Online first

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

2. 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, dynamics
Categories:37B99, 54H20, 54E52

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 dynamics

4. 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 space
Categories:37D20, 37B10

5. 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

6. 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 Kähler 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

7. 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 dynamics

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