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Search: MSC category 54H20 ( Topological dynamics [See also 28Dxx, 37Bxx] )

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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 Online first

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 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 Erdős), 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

4. CJM 2002 (vol 54 pp. 1187)

Cobo, Milton; Gutierrez, Carlos; Llibre, Jaume
On the Injectivity of $C^1$ Maps of the Real Plane
Let $X\colon\mathbb{R}^2\to\mathbb{R}^2$ be a $C^1$ map. Denote by $\Spec(X)$ the set of (complex) eigenvalues of $\DX_p$ when $p$ varies in $\mathbb{R}^2$. If there exists $\epsilon >0$ such that $\Spec(X)\cap(-\epsilon,\epsilon)=\emptyset$, then $X$ is injective. Some applications of this result to the real Keller Jacobian conjecture are discussed.

Categories:34D05, 54H20, 58F10, 58F21

5. CJM 2001 (vol 53 pp. 325)

Matui, Hiroki
Ext and OrderExt Classes of Certain Automorphisms of $C^*$-Algebras Arising from Cantor Minimal Systems
Giordano, Putnam and Skau showed that the transformation group $C^*$-algebra arising from a Cantor minimal system is an $AT$-algebra, and classified it by its $K$-theory. For approximately inner automorphisms that preserve $C(X)$, we will determine their classes in the Ext and OrderExt groups, and introduce a new invariant for the closure of the topological full group. We will also prove that every automorphism in the kernel of the homomorphism into the Ext group is homotopic to an inner automorphism, which extends Kishimoto's result.

Categories:46L40, 46L80, 54H20

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