1,377 research outputs found
Shape of matchbox manifolds
In this work, we develop shape expansions of minimal matchbox manifolds
without holonomy, in terms of branched manifolds formed from their leaves. Our
approach is based on the method of coding the holonomy groups for the foliated
spaces, to define leafwise regions which are transversely stable and are
adapted to the foliation dynamics. Approximations are obtained by collapsing
appropriately chosen neighborhoods onto these regions along a "transverse
Cantor foliation". The existence of the "transverse Cantor foliation" allows us
to generalize standard techniques known for Euclidean and fibered cases to
arbitrary matchbox manifolds with Riemannian leaf geometry and without
holonomy. The transverse Cantor foliations used here are constructed by purely
intrinsic and topological means, as we do not assume that our matchbox
manifolds are embedded into a smooth foliated manifold, or a smooth manifold.Comment: 36 pages. Revision of the earlier version: introduction is rewritten.
Accepted to a special issue of Indagationes Mathematica
The Normal Form Theorem around Poisson Transversals
We prove a normal form theorem for Poisson structures around Poisson
transversals (also called cosymplectic submanifolds), which simultaneously
generalizes Weinstein's symplectic neighborhood theorem from symplectic
geometry and Weinstein's splitting theorem. Our approach turns out to be
essentially canonical, and as a byproduct, we obtain an equivariant version of
the latter theorem.Comment: 15 pages; v2: the title was changed; v3: proof of Lemma 2 was
include
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
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