Growth and Structure of Equicontinuous Foliated Spaces

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.