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2024 | OriginalPaper | Buchkapitel

Growth and Structure of Equicontinuous Foliated Spaces

verfasst von : Manuel F. Moreira Galicia

Erschienen in: Differential Geometric Structures and Applications

Verlag: Springer Nature Switzerland

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Abstract

Molino’s description of Riemannian foliations on compact manifolds extends to compact equicontinuous foliated spaces as developed by Àlvarez Lòpez and Manuel Moreira. This extension is particularly considered when leaves are densely packed, and the pseudogroup exhibits strong quasi-analytic behavior. Notably, this extension leads to the establishment of an association with a structural local group within such a foliated space. Application of this framework results in a partial generalization of Carrière and Breuillard-Gelander’s results, creating a connection between the structural local group and the growth of leaves. Additionally, we present an illustrative examples, as well as a scenario with zero topological codimension. In this context, instances of weak solenoids embodying this characteristic have been previously established by Dyer, Hurder, and Lukina.

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Vorheriges Kapitel Minimal Unit Vector Fields on Oscillator Groups
Nächstes Kapitel Lichnerowicz-Type Laplacians in the Bochner Technique
Fußnoten
1
This is usually called étalé morphism. We simply call it morphism because no other type of morphism will be considered here.
 
2
Recall that a space is called Polish if it is separable and completely metrizable.
 
3
A local group is said to have no small subgroups when some neighborhood of the identity element contains no nontrivial subgroup.
 
4
The notation will be simplified by using, for instance, \(\{T_i,d_i\}\) instead of \(\{(T_i,d_i)\}\).
 
5
In fact, it induces a uniformity. We could even use any uniformity to define equicontinuity, but such generality will not be used here.
 
6
This adverb, used in [4], will be omitted for the sake of simplicity.
 
7
A shrinking of an open cover \(\{U_i\}\) of a space X is an open cover \(\{U'_i\}\) of X, with the same index set, such that \(\overline{U'_i}\subset U_i\) for all i. Similarly, if \(\{U_i\}\) is a cover of a subset \(A\subset X\) by open subsets of X, a shrinking of \(\{U_i\}\), as a cover of A by open subsets of X, is a cover \(\{U'_i\}\) of A by open subsets of X, with the same index set, so that \(\overline{U'_i}\subset U_i\) for all i.
 
8
Usually, growth types are defined by using non-decreasing functions \({\mathbb {Z}}^+\rightarrow [0,\infty )\), but non-decreasing functions \([0,\infty )\rightarrow [0,\infty )\) give rise to an equivalent concept.
 
9
Recall that a continuum is a non-empty compact connected metrizable space.
 
10
The logical problems of this definition can be avoided because any complete connected Riemannian manifold is equipotent to some subset of \({\mathbb R}\).
 
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Metadaten
Titel
Growth and Structure of Equicontinuous Foliated Spaces
verfasst von
Manuel F. Moreira Galicia
Copyright-Jahr
2024
Buch
Differential Geometric Structures and Applications

Print ISBN: 978-3-031-50585-0

Electronic ISBN: 978-3-031-50586-7

Copyright-Jahr: 2024

DOI
https://doi.org/10.1007/978-3-031-50586-7_7

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