Christian Skau

Substitutional dynamical systems, Bratteli diagrams and dimension groups

The present paper explores substitution minimal systems and their relation to stationary Bratteli diagrams and stationary dimension groups. The constructions involved are algorithmic and explicit, and render an effective method to compute an invariant of (ordered)

Full groups of Cantor minimal systems

K

Orbit equivalence for Cantor minimal ℤd-systems

-theoretic nature for these systems. This new invariant is independent of spectral invariants which have previously been extensively studied. Before we state the main results we give some background.

Orbit equivalence for Cantor minimal

We associate different types of full groups to Cantor minimal systems. We show how these various groups (as abstract groups) are complete invariants for orbit equivalence, strong orbit equivalence and flip conjugacy, respectively. Furthermore, we introduce a group homomorphism, the socalled mod map, from the normalizers of the various full groups to the automorphism groups of the (ordered)K0-groups, which are associated to the Cantor minimal systems. We show how this in turn is related to the automorphisms of the associatedC*-crossed products. Our results are analogues in the topological dynamical setting of results obtained by Dye, Connes-Krieger and Hamachi-Osikawa in measurable dynamics.

\mathbb{Z}^{2}

We show that every minimal action of any finitely generated abelian group on the Cantor set is (topologically) orbit equivalent to an AF relation. As a consequence, this extends the classification up to orbit equivalence of minimal dynamical systems on the Cantor set to include AF relations and ℤd-actions.

-systems

We show that every minimal action of any finitely generated abelian group on the Cantor set is (topologically) orbit equivalent to an AF relation. As a consequence, this extends the classification up to orbit equivalence of minimal dynamical systems on the Cantor set to include AF relations and ℤ d -actions.

The absorption theorem for affable equivalence relations

We prove a result about extension of a minimal AF-equivalence relation R on the Cantor set X, the extension being ‘small’ in the sense that we modify R on a thin closed subset Y of X. We show that the resulting extended equivalence relation S is orbit equivalent to the original R, and so, in particular, S is affable. Even in the simplest case—when Y is a finite set—this result is highly non-trivial. The result itself—called the absorption theorem—is a powerful and crucial tool for the study of the orbit structure of minimal Zn-actions on ∗Supported in part by a grant from NSERC, Canada †Supported in part by a grant from the Japan Society for the Promotion of Science ‡Supported in part by a grant from NSERC, Canada §Supported in part by the Norwegian Research Council

The Orbit Structure of Cantor Minimal Z2-Systems

In 1959, H. Dye ([D1]) introduced the notion of orbit equivalence and proved that any two ergodic finite measure preserving transformations on a Lebesgue space are orbit equivalent. In [D2], he had also conjectured that an arbitrary ergodic action of a discrete amenable group is orbit equivalent to a Z-action. This conjecture was proved by Ornstein and Weiss in [OW]. The most general case was proved by Connes, Feldman and Weiss ([CFW]) by establishing that an amenable non-singular countable equivalence relation R can be generated by a single transformation, or equivalently, is hyperfinite, i.e., R is up to a null set, a countable increasing union of finite equivalence relations. For the Borel case, Weiss ([W]) proved that actions of Z are (orbit equivalent to) hyperfinite Borel equivalence relations, whose classification was obtained by Dougherty, Jackson and Kechris ([DJK]). It is not yet known if an arbitrary Borel action of a discrete amenable group is orbit equivalent to a Z-action. Our main interest in this report is the case of a free minimal continuous action φ of Z on a Cantor set (i.e., a compact totally disconnected metric space with no isolated points). However, let us begin with a more general group action and consider a free action φ of a countable discrete group on a compact metric space X (i.e., for every g ∈ G , φ(g) ∈ Homeo (X), and φ(g)x = x for some x ∈ X if and only if g = id). Recall that the action φ is minimal if the φ-orbit of every point of X is dense in X.

COCYCLES FOR CANTOR MINIMAL ℤd-SYSTEMS

We consider a minimal, free action, φ, of the group ℤd on the Cantor set X, for d ≥ 1. We introduce the notion of small positive cocycles for such an action. We show that the existence of such cocycles allows the construction of finite Kakutani–Rohlin approximations to the action. In the case, d = 1, small positive cocycles always exist and the approximations provide the basis for the Bratteli–Vershik model for a minimal homeomorphism of X. Finally, we consider two classes of examples when d = 2 and show that such cocycles exist in both.

The lord of the numbers, Atle Selberg. On his life and mathematics

The renowned Norwegian mathematician Atle Selberg died on 6 August, 2007, in his home in Princeton. He was one of the giants of the twentieth centuries mathematics. His contributions to mathematics are so deep and original that his name will always be an important part of the history of mathematics. His special field was number theory in a broad sense. Atle Selberg was born on June 14, 1917, in Langesund, Norway. He grew up near Bergen and went to high school at Gjøvik. His father was a high school teacher with a doctoral degree in mathematics, and two of his older brothers, Henrik and Sigmund, became professors of mathematics in Norway. He was studying mathematics at the university level at the age of 12. When he was 15 he published a little note in Norsk Matematisk Tidsskrift. He studied at the University of Oslo where he obtained the Cand. real. degree in 1939, and in the autumn of 1943 he defended his thesis, which was about the Riemann Hypothesis. At that time there was little numerical evidence supporting the Riemann Hypothesis. He got the idea of studying the zeros of the Riemann zeta-function as a kind of moment problem, and this led to his famous estimate of the number of zeros. From this it followed that a positive fraction of the zeros must lie on the critical line. This result led to great international recognition. When Carl Ludwig Siegel, who had stayed in the United States, asked Harald Bohr what had happened in mathematics in Europe during the war, Bohr answered with one word: Selberg. During the summer of 1946, Selberg realized that his work on the Riemann zeta function could be applied to estimate the number of primes in an interval. This was the beginning of the development leading to the famous Selberg sieve-method.