the present article, and I would like to thank my collaborators Rostislav Grigorchuk and Tatiana Nagnibeda for the gentle insistence with which they encouraged my work.

Structure of the ergodic components and recurrence properties
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Lebesgue spaces {#sec:L}
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We begin by recalling the basic properties of the *Lebesgue measure spaces* introduced by Rokhlin, see [@Rohlin52; @Cornfeld-Fomin-Sinai82]. Measure-theoretically these are the probability spaces such that their non-atomic part is isomorphic to an interval with the Lebesgue measure on it. Thus, any Lebesgue space is uniquely characterized by its *signature* ${\sigma}=({\sigma}_0;{\sigma}_1,{\sigma}_2,\dots)$, where ${\sigma}_0$ is the total mass of the non-atomic part, and ${\sigma}_1\ge{\sigma}_2\ge\dots$ is the ordered sequence of the values of its atoms (extended by zeroes if the set of atoms is finite or empty). There is also an intrinsic definition of Lebesgue spaces based on their separability properties. However, for applications it is usually enough to know that any *Polish topological space* (i.e., separable, metrizable, complete) endowed with a Borel probability measure is a Lebesgue measure space. We shall follow the standard measure theoretical convention:

*Unless otherwise specified, all the identities, properties etc. related to measure spaces will be understood [mod 0]{} (i.e., up to null sets)*. In particular, all 