81 research outputs found
On right conjugacy closed loops of twice prime order
The right conjugacy closed loops of order 2p, where p is an odd prime, are
classified up to isomorphism.Comment: Clarified definitions, added some remarks and a tabl
Loops and quasigroups: Aspects of current work and prospects for the future
summary:This paper gives a brief survey of certain recently developing aspects of the study of loops and quasigroups, focussing on some of the areas that appear to exhibit the best prospects for subsequent research and for applications both inside and outside mathematics
Lipshitz matchbox manifolds
A matchbox manifold is a connected, compact foliated space with totally
disconnected transversals; or in other notation, a generalized lamination. It
is said to be Lipschitz if there exists a metric on its transversals for which
the holonomy maps are Lipschitz. Examples of Lipschitz matchbox manifolds
include the exceptional minimal sets for -foliations of compact manifolds,
tiling spaces, the classical solenoids, and the weak solenoids of McCord and
Schori, among others. We address the question: When does a Lipschitz matchbox
manifold admit an embedding as a minimal set for a smooth dynamical system, or
more generally for as an exceptional minimal set for a -foliation of a
smooth manifold? We gives examples which do embed, and develop criteria for
showing when they do not embed, and give examples. We also discuss the
classification theory for Lipschitz weak solenoids.Comment: The paper has been significantly revised, with several proofs
correcte
Classifying matchbox manifolds
Matchbox manifolds are foliated spaces with totally disconnected
transversals. Two matchbox manifolds which are homeomorphic have return
equivalent dynamics, so that invariants of return equivalence can be applied to
distinguish non-homeomorphic matchbox manifolds. In this work we study the
problem of showing the converse implication: when does return equivalence imply
homeomorphism? For the class of weak solenoidal matchbox manifolds, we show
that if the base manifolds satisfy a strong form of the Borel Conjecture, then
return equivalence for the dynamics of their foliations implies the total
spaces are homeomorphic. In particular, we show that two equicontinuous
\mT^n--like matchbox manifolds of the same dimension are homeomorphic if and
only if their corresponding restricted pseudogroups are return equivalent. At
the same time, we show that these results cannot be extended to include the
"\emph{adic}-surfaces", which are a class of weak solenoids fibering over a
closed surface of genus 2.Comment: This work is an extensive revision of the previous version on the
arXi
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