Bratteli-Vershik models for Cantor minimal systems associated to interval exchange transformations

Abstract

We construct Bratteli-Vershik models for minimal interval exchange transformations. We use this to show that the interval exchange transformations over quadratic fields, recently studied by Boshernitzan and Carroll, actually are (conjugate to) substitution minimal systems. We also prove a partial converse to this. Furthermore, these systems are orbit equivalent to Sturmian systems.

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

“…Here we suggest constructions of universal graphs on which every automorphism can be realized as an adic shift. The study of specific automorphisms has already begun (see the survey [1], and also [2]), but, according to the conclusion of the paper [16] (see also below), the problem of classification of combinatorially definite filtrations is tame, i.e., there is a manageable space of classes, or orbits, or complete metric invariants of such filtrations (see the definition of combinatorial schemes of hierarchies below). This gives a chance to obtain a reasonable combinatorial classification of measure-preserving automorphisms.…”

Infinite type flat surface models of ergodic systems

“…Let S i k ⊂ R i k , S j k ⊂ R j k be two components of R k (S k ) and suppose that a left/right edge of R i k is identified to some right/left edge of R j k . We claim that we can connect S i k with S j k by a path γ k i,j : [0, 1] → S k (B, w ± , ≤ r,s ) which satisfies (25) min…”

“…Therefore, for any two sets S i k and S j k , if we can connect them by a path γ, using (25) and (26) {w ± (v)} and therefore justified our choice for δ t k in (19) for (3). We have showed why (21).…”

“…What we do in this paper is very similar in spirit, and is a continuation and generalization of the point of view introduced in [14], but our goals are different: whereas [14] are mostly interested in studying limit distributions of translation flows on compact surfaces and [12,13] is concerned with limit theorems of flows over symbolic systems, we are interested in developing a theory connecting symbolic systems (Vershik automorphisms) obtained from bi-infinite Bratteli diagrams with flat surfaces of finite and infinite genus. It is known that any compact flat surface can be recovered through a Bratteli diagram of a very specific type which is dictated from the Rauzy-Veech induction [25]. The flat surfaces that arise via the construction described here are, in most cases, of infinite topological type.…”

Предельные Теоремы Для Специальных Потоков Над Преобразованиями Вершика

“…В случае перекладываний отрезков такая последова-тельность башен получается, например, при индукции Рози-Вича; в результа-те мы получаем явное представление минимального перекладывания отрезков в виде автоморфизма Вершика (см. [25]). В следующем п.…”