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Losing Hausdorff dimension while generating pseudogroups

100%
Walczak P. G.
Fundamenta Mathematicae
|
1996
|
tom 149
|
nr 3
211-237
EN
Considering different finite sets of maps generating a pseudogroup G of locally Lipschitz homeomorphisms between open subsets of a compact metric space X we arrive at a notion of a Hausdorff dimension $dim_H G$ of G. Since $dim_H G ≤ dim_H X$, the dimension loss $dl_HG = dim_HX - dim_H G$ can be considered as a "topological price" one has to pay to generate G. We collect some properties of $dim_H$ and $dl_H$ (for example, both of them are invariant under Lipschitz isomorphisms of pseudogroups) and we either estimate or calculate $dim_HG$ for pseudogroups arising from classical dynamical systems, group actions, foliations, etc.
2
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Jacobi operator for leaf geodesics

100%
Walczak P. G.
Colloquium Mathematicum
|
1993
|
tom 65
|
nr 2
213-226
3
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Transverse Hausdorff dimension of codim-1 C2-foliations

64%
Inaba T., Walczak P. G.
Fundamenta Mathematicae
|
1996
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tom 149
|
nr 3
239-244
EN
The Hausdorff dimension of the holonomy pseudogroup of a codimension-one foliation ℱ is shown to coincide with the Hausdorff dimension of the space of compact leaves (traced on a complete transversal) when ℱ is non-minimal, and to be equal to zero when ℱ is minimal with non-trivial leaf holonomy.
4
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On the energy of unit vector fields with isolated singularities

64%
Brito F. G. B., Walczak P. G.
Annales Polonici Mathematici
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2000
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tom 73
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nr 3
269-274
EN
We consider the energy of a unit vector field defined on a compact Riemannian manifold M except at finitely many points. We obtain an estimate of the energy from below which appears to be sharp when M is a sphere of dimension >3. In this case, the minimum of energy is attained if and only if the vector field is totally geodesic with two singularities situated at two antipodal points (at the 'south and north pole').
5
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An integral formula for a Riemannian manifold with two orthogonal complementary distributions

63%
Walczak P. G.
Colloquium Mathematicum
|
1989-1990
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tom 58
|
nr 2
243-252
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