A matchbox manifold is a foliated space with totally disconnected transversals, and an equicontinuous matchbox manifold is the generalization of Riemannian foliations for smooth manifolds in this context. In this paper, we develop the Molino theory for all equicontinuous matchbox manifolds. Our work extends the Molino theory developed in the work of Alvarez Lopez and Moreira Galicia which required the hypothesis that the holonomy actions for these spaces satisfy the strong quasi-analyticity condition. The methods of this paper are based on the authors previous works on the structure of weak solenoids, and provide many new properties of the Molino theory for the case of totally disconnected transversals, and examples to illustrate these properties. In particular, we show that the Molino space need not be uniquely well-defined, unless the global holonomy dynamical system is tame, a notion defined in this work. We show that examples in the literature for the theory of weak solenoids provide examples for which the strong quasi-analytic condition fails. Of particular interest is a new class of examples of equicontinuous minimal Cantor actions by finitely generated groups, whose construction relies on a result of Lubotzky. These examples have non-trivial Molino sequences, and other interesting properties.
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