22 research outputs found
Lattice packings with gap defects are not completely saturated
We show that a honeycomb circle packing in with a linear gap defect
cannot be completely saturated, no matter how narrow the gap is. The result is
motivated by an open problem of G. Fejes T\'oth, G. Kuperberg, and W.
Kuperberg, which asks whether of a honeycomb circle packing with a linear shift
defect is completely saturated. We also show that an fcc sphere packing in
with a planar gap defect is also not completely saturated
Bihomogeneity and Menger manifolds
For every triple of integers a, b, and c, such that a>O, b>0, and c>1, there
is a homogeneous, non-bihomogeneous continuum whose every point has a
neighborhood homeomorphic the Cartesian product of three Menger compacta m^a,
m^b, and m^c. In particular, there is a homogeneous, non-bihomogeneous, Peano
continuum of covering dimension four.Comment: 9 page
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Generalized counterexamples to the Seifert conjecture
Using the theory of plugs and the self-insertion construction due to the second
author, we prove that a foliation of any codimension of any manifold can be modified in a
real analytic or piecewise-linear fashion so that all minimal sets have codimension 1. In
particular, the 3-sphere S^3 has a real analytic dynamical system such that all limit sets
are 2-dimensional. We also prove that a 1-dimensional foliation of a manifold of dimension
at least 3 can be modified in a piecewise-linear fashion so that there are no closed leaves
but all minimal sets are 1-dimensional. These theorems provide new counterexamples to the
Seifert conjecture, which asserts that every dynamical system on S^3 with no singular
points has a periodic trajectory
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Lattice packings with gap defects are not completely saturated
We show that a honeycomb circle packing in with a linear gap defect cannot
be completely saturated, no matter how narrow the gap is. The result is motivated by an
open problem of G. Fejes T\'oth, G. Kuperberg, and W. Kuperberg, which asks whether of a
honeycomb circle packing with a linear shift defect is completely saturated. We also show
that an fcc sphere packing in with a planar gap defect is also not completely
saturated