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1. CJM Online first
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 |
2. CJM 2013 (vol 65 pp. 1287)
$K$-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 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 |
3. CJM 2009 (vol 61 pp. 708)
Regular Homeomorphisms of Finite Order on Countable Spaces We present a structure theorem for a broad class of homeomorphisms of
finite order on countable zero dimensional spaces. As applications we
show the following.
\begin{compactenum}[\rm(a)]
\item Every countable nondiscrete topological group not containing an
open Boolean subgroup can be partitioned into infinitely many dense
subsets.
\item If $G$ is a countably infinite Abelian group with finitely many
elements of order $2$ and $\beta G$ is the Stone--\v Cech
compactification of $G$ as a discrete semigroup, then for every
idempotent $p\in\beta G\setminus\{0\}$, the subset
$\{p,-p\}\subset\beta G$ generates algebraically the free product of
one-element semigroups $\{p\}$ and~$\{-p\}$.
\end{compactenum}
Keywords:Homeomorphism, homogeneous space, topological group, resolvability, Stone-\v Cech compactification Categories:22A30, 54H11, 20M15, 54A05 |
4. CJM 2006 (vol 58 pp. 529)
On the Group of Homeomorphisms of the Real Line That Map the Pseudoboundary Onto Itself In this paper we primarily consider two natural subgroups of the autohomeomorphism group of the real
line $\R$, endowed with the compact-open topology. First, we prove that the subgroup of
homeomorphisms that map the set of rational numbers $\Q$ onto itself
is homeomorphic to the infinite power of $\Q$ with
the product topology. Secondly, the group consisting of homeomorphisms that map the pseudoboundary
onto itself is shown to be homeomorphic to the hyperspace of nonempty compact subsets of $\Q$ with
the Vietoris topology. We obtain similar results for the Cantor set but we also prove that these
results do not extend to $\R^n$ for $n\ge 2$, by linking the groups in question with Erd\H os
space.
Keywords:homeomorphism group, real line, countable dense set, pseudoboundary, Erd\H{o}s space, hyperspace Category:57S05 |