Combinatorial Invariants of Metric Filtrations and Automorphisms; the Universal Adic Graph

On a Universal Borel Adic Space

“…As in the metric case (see [10]), every basic Borel filtration of a separable metric space can be colored so as to become combinatorially definite. Proposition 1.…”

“…The construction suggested in [10] allows one to realize combinatorially definite (colored) basic filtrations as tail filtrations on graded graphs endowed with an adic structure. It can be carried over to the Borel case without changes.…”

“…The corollaries obtained in this paper and in [10] concern the theory of uniform approximation of actions of the group Z, in particular, a new method of encoding automorphisms via filtrations.…”

“…In [6,7], the first author proved that every ergodic automorphism of a Lebesgue space has an adic realization, i.e., is isomorphic to the adic shift on the path space of some graded graph equipped with a central measure. In [10], it is proved that for such a graph one can always take the so-called uniadic graph UA (see Sec. 4), varying only a central measure on its path space.…”

“…The first one consists in obtaining a Borel version of Rokhlin's lemma (see [1]). The second idea is to iterate a weakened version of Rokhlin's lemma in order to construct a basic Borel filtration and an adic realization of the automorphism (see [6,7] and [10]). We will prove a weakened Borel version of Rokhlin's lemma (Lemma 2), iterate it to construct a basic Borel filtration of the automorphism (Theorem 2), and prove Theorem 1.…”