Full groups, flip conjugacy, and orbit equivalence of Cantor minimal systems

In the paper, we consider the full group $[\phi]$ and topological full group $[[\phi]]$ of a Cantor minimal system (X,\f). We prove that the commutator subgroups D([\f]) and D([[\f]]) are simple and show that the groups D([\f]) and D([[\f]]) completely determine the class of orbit equivalence and flip conjugacy of \f, respectively. These results improve the classification found in \cite{gps:1999}. As a corollary of the technique used, we establish the fact that \f can be written as a product of three involutions from [\f].Comment: 17 pages, references added, some typos fixe

Perfect orderings on Bratteli diagrams II: general Bratteli diagrams

We continue our study of orderings on Bratteli diagrams started in previous work, joint with Jan Kwiatkowski, where Bratteli diagrams of finite rank were considered. We extend the notions of languages, permutations (called correspondences in this paper), skeletons and associated graphs to the case of general Bratteli diagrams, and show their relevance to the study of perfect orderings: those that support Vershik maps; in particular, perfect orderings with several extremal paths. A perfect ordering comes equipped with a skeleton and a correspondence, and conversely, given a skeleton and correspondence, we describe explicitly how to construct perfect orderings, by showing that paths in the associated directed graphs determine the language of the order. We describe an explicit algorithmic method to create perfect orderings on Bratteli diagrams based on the study of certain relations between the entries of the diagram's incidence matrices and properties of the associated graphs, with the latter relations characterizing diagrams which support perfect orderings. Also, we apply the notions of skeletons and associated graphs, to give a new combinatorial proof of the fact that diagrams supporting perfect orderings with k maximal paths have a direct sum of k-1 copies of the integers contained in their infinitesimal subgroup. Under certain conditions, we show that a similar result holds if the diagram supports countably many maximal paths. Our results are illustrated by numerous examples.Comment: 32 pages, 6 figures. This version incorporates the referee's remarks. arXiv admin note: text overlap with arXiv:1204.162

Measures on Cantor sets: the good, the ugly, the bad

We translate Akin's notion of {\it good} (and related concepts) from measures on Cantor sets to traces on dimension groups, and particularly for invariant measures of minimal homeomorphisms (and their corresponding simple dimension groups), this yields characterizations and examples, which translate back to the original context. Good traces on a simple dimension group are characterized by their kernel having dense image in their annihilating set of affine functions on the trace space; this makes it possible to construct many examples with seemingly paradoxical properties. In order to study the related property of {\it refinability,} we consider goodness for sets of measures (traces on dimension groups), and obtain partial characterizations in terms of (special) convex subsets of Choquet simplices. These notions also very closely related to unperforation of quotients of dimension groups by convex subgroups (that are not order ideals), and we give partial characterizations. Numerous examples illustrate the results

Monopoles, dipoles, and harmonic functions on Bratteli diagrams

In our study of electrical networks we develop two themes: finding explicit formulas for special classes of functions defined on the vertices of a transient network, namely monopoles, dipoles, and harmonic functions. Secondly, our interest is focused on the properties of electrical networks supported on Bratteli diagrams. We show that the structure of Bratteli diagrams allows one to describe algorithmically harmonic functions as well as monopoles and dipoles. We also discuss some special classes of Bratteli diagrams (stationary, Pascal, trees), and we give conditions under which the harmonic functions defined on these diagrams have finite energy.Comment: 7 figure

Laplace operators in finite energy and dissipation spaces

Recent applications of large network models to machine learning, and to neural network suggest a need for a systematic study of the general correspondence, (i) discrete vs (ii) continuous. Even if the starting point is (i), limit considerations lead to (ii), or, more precisely, to a measure theoretic framework which we make precise. Our motivation derives from graph analysis, e.g., studies of (infinite) electrical networks of resistors, but our focus will be (ii), i.e., the measure theoretic setting. In electrical networks of resistors, one considers pairs (of typically countably infinite), sets $V$ (vertices), $E$ (edges) a suitable subset of $V \times V$, and prescribed positive symmetric functions $c$ on $E$ . A conductance function $c$ is defined on $E$ (edges), or on $V \times V$, but with $E$ as its support. From an initial triple $(V, E, c)$ , one gets graph-Laplacians, generalized Dirichlet spaces (also called energy Hilbert spaces), dipoles, relative reproducing kernel-theory, dissipation spaces, reversible Markov chains, and more. Our main results include: spectral theory and Green's functions for measure theoretic graph-Laplace operators; the theory of reproducing kernel Hilbert spaces related to Laplace operators; a rigorous analysis of the Laplacian on Borel equivalence relations; a new decomposition theory; irreducibility criteria; dynamical systems governed by endomorphisms and measurable fields; orbit equivalence criteria; and path-space measures and induced dissipation Hilbert spaces. We consider several applications of our results to other fields such as machine learning problems, reproducing kernel Hilbert spaces, Gaussian and determinantal processes, and joinings.Comment: 70 page

Topologies on the group of Borel automorphisms of a standard Borel space

The paper is devoted to the study of topologies on the group Aut(X,B) of all Borel automorphisms of a standard Borel space $(X, B)$. Several topologies are introduced and all possible relations between them are found. One of these topologies, $\tau$, is a direct analogue of the uniform topology widely used in ergodic theory. We consider the most natural subsets of $Aut(X, B)$ and find their closures. In particular, we describe closures of subsets formed by odometers, periodic, aperiodic, incompressible, and smooth automorphisms with respect to the defined topologies. It is proved that the set of periodic Borel automorphisms is dense in $Aut(X, B)$ (Rokhlin lemma) with respect to $\tau$. It is shown that the $\tau$-closure of odometers (and of rank 1 Borel automorphisms) coincides with the set of all aperiodic automorphisms. For every aperiodic automorphism $T\in Aut(X, B)$, the concept of a Borel-Bratteli diagram is defined and studied. It is proved that every aperiodic Borel automorphism $T$ is isomorphic to the Vershik transformation acting on the space of infinite paths of an ordered Borel-Bratteli diagram. Several applications of this result are given.Comment: 53 page

The Rokhlin lemma for homeomorphisms of a Cantor set

For a Cantor set $X$, let $Homeo(X)$ denote the group of all homeomorphisms of $X$. The main result of this note is the following theorem. Let $T\in Homeo(X)$ be an aperiodic homeomorphism, let $\mu_1,\mu_2,...,\mu_k$ be Borel probability measures on $X$, \e> 0, and $n \ge 2$. Then there exists a clopen set $E\subset X$ such that the sets $E,TE,..., T^{n-1}E$ are disjoint and \mu_i(E\cup TE\cup...\cup T^{n-1}E) > 1 - \e, i= 1,...,k. Several corollaries of this result are given. In particular, it is proved that for any aperiodic $T\in Homeo(X)$ the set of all homeomorphisms conjugate to $T$ is dense in the set of aperiodic homeomorphisms.Comment: 9 pages. Proc. Ams, to appea

Topologies on the group of homeomorphisms of a Cantor set

Let $Homeo(\Omega)$ be the group of all homeomorphisms of a Cantor set $\Omega$. We study topological properties of $Homeo(\Omega)$ and its subsets with respect to the uniform $(\tau)$ and weak $(\tau_w)$ topologies. The classes of odometers and periodic, aperiodic, minimal, rank 1 homeomorphisms are considered and the closures of those classes in $\tau$ and $\tau_w$ are found.Comment: 33 page

Finite Rank Bratteli Diagrams: Structure of Invariant Measures

We consider Bratteli diagrams of finite rank (not necessarily simple) and ergodic invariant measures with respect to the cofinal equivalence relation on their path spaces. It is shown that every ergodic invariant measure (finite or "regular" infinite) is obtained by an extension from a simple subdiagram. We further investigate quantitative properties of these measures, which are mainly determined by the asymptotic behavior of products of incidence matrices. A number of sufficient conditions for unique ergodicity are obtained. One of these is a condition of exact finite rank, which parallels a similar notion in measurable dynamics. Several examples illustrate the broad range of possible behavior of finite type diagrams and invariant measures on them. We then prove that the Vershik map on the path space of an exact finite rank diagram cannot be strongly mixing, independent of the ordering. On the other hand, for the so-called "consecutive" ordering, the Vershik map is not strongly mixing on all finite rank diagrams.Comment: 49 pages. A reworked version of the pape

Cohomology of hyperfinite Borel actions

We study cocycles of countable groups $\Gamma$ of Borel automorphisms of a standard Borel space $(X, \mathcal{B})$ taking values in a locally compact second countable group $G$. We prove that for a hyperfinite group $\Gamma$ the subgroup of coboundaries is dense in the group of cocycles. We describe all Borel cocycles of the $2$-odometer and show that any such cocycle is cohomologous to a cocycle with values in a countable dense subgroup $H$ of $G$. We also provide a Borel version of Gottschalk-Hedlund theorem