Short proofs of theorems of Malyutin and Margulis

The Ghys-Margulis alternative asserts that a subgroup $G$ of homeomorphisms of the circle which does not contain a free subgroup on two generators must admit an invariant probability measure. Malyutin's theorem classifies minimal actions of $G$. We present a short proof of Malyutin's theorem and then deduce Margulis' theorem which confirms the G-M alternative. The basic ideas are borrowed from the original work of Malyutin but the use of the apparatus of the enveloping semigroup enables us to shorten the proof considerably

RP[d]RP^{[d]} is an equivalence relation: An enveloping semigroup proof

We present a purely enveloping semigroup proof of a theorem of Shao and Ye which asserts that for an abelian group $T$, a minimal flow $(X,T)$ and any integer $d \ge 1$, the regional proximal relation of order $d$ is an equivalence relation.Comment: just a few minor correction

The structure of tame minimal dynamical systems for general groups

We use the structure theory of minimal dynamical systems to show that, for a general group $\Gamma$, a tame, metric, minimal dynamical system $(X, \Gamma)$ has the following structure: \begin{equation*} \xymatrix {& \tilde{X} \ar[dd]_\pi \ar[dl]_\eta & X^* \ar[l]_-{\theta^*} \ar[d]^{\iota} \ar@/^2pc/@{>}^{\pi^*}[dd]\\ X & & Z \ar[d]^\sigma\\ & Y & Y^* \ar[l]^\theta } \end{equation*} Here (i) $\tilde{X}$ is a metric minimal and tame system (ii) $\eta$ is a strongly proximal extension, (iii) $Y$ is a strongly proximal system, (iv) $\pi$ is a point distal and RIM extension with unique section, (v) $\theta$, $\theta^*$ and $\iota$ are almost one-to-one extensions, and (vi) $\sigma$ is an isometric extension. When the map $\pi$ is also open this diagram reduces to \begin{equation*} \xymatrix {& \tilde{X} \ar[dl]_\eta \ar[d]^{\iota} \ar@/^2pc/@{>}^\pi[dd]\\ X & Z \ar[d]^\sigma\\ & Y } \end{equation*} In general the presence of the strongly proximal extension $\eta$ is unavoidable. If the system $(X, \Gamma)$ admits an invariant measure $\mu$ then $Y$ is trivial and $X = \tilde{X}$ is an almost automorphic system; i.e. $X \overset{\iota}{\to} Z$, where $\iota$ is an almost one-to-one extension and $Z$ is equicontinuous. Moreover, $\mu$ is unique and $\iota$ is a measure theoretical isomorphism $\iota : (X,\mu, \Gamma) \to (Z, \lambda, \Gamma)$, with $\lambda$ the Haar measure on $Z$. Thus, this is always the case when $\Gamma$ is amenable.Comment: 27 pages; to appear in Invent. Math. arXiv admin note: substantial text overlap with arXiv:math/060950

On two problems concerning topological centers

Let G be an infinite discrete group and bG its Cech-Stone compactification. Using the well known fact that a free ultrafilter on an infinite set is nonmeasurable, we show that for each element p of the remainder bG G, left multiplication L_p:bG \to bG is not Borel measurable. Next assume that G is abelian. Let D \subset \ell^\infty(G)$ denote the subalgebra of distal functions on G and let G^D denote the corresponding universal distal (right topological group) compactification of G. Our second result is that the topological center of G^D (i.e. the set of p in G^D for which L_p:G^D \to G^D is a continuous map) is the same as the algebraic center and that for G=Z (the group of integers) this center coincides with the canonical image of G in G^D

Representations of dynamical systems on Banach spaces not containing l1l_1

For a topological group G, we show that a compact metric G-space is tame if and only if it can be linearly represented on a separable Banach space which does not contain an isomorphic copy of $l_1$ (we call such Banach spaces, Rosenthal spaces). With this goal in mind we study tame dynamical systems and their representations on Banach spaces.Comment: 27 pages, revised version, to appear in Trans. AM

A generic distal tower of arbitrary countable height over an arbitrary infinite ergodic system

We show the existence, over an arbitrary infinite ergodic $\mathbb{Z}$-dynamical system, of a generic ergodic relatively distal extension of arbitrary countable rank and arbitrary infinite compact extending groups (or more generally, infinite quotients of compact groups) in its canonical distal tower.Comment: Improved presentation and corrected misprint

On Hilbert dynamical systems

Returning to a classical question in Harmonic Analysis we strengthen an old result of Walter Rudin. We show that there exists a weakly almost periodic function on the group of integers Z which is not in the norm-closure of the algebra B(Z) of Fourier-Stieltjes transforms of measures on the circle, the dual group of Z, and which is recurrent. We also show that there is a Polish monothetic group which is reflexively but not Hilbert representable

Uniformly recurrent subgroups

We define the notion of uniformly recurrent subgroup, URS in short, which is a topological analog of the notion of invariant random subgroup (IRS), introduced in a work of M. Abert, Y. Glasner and B. Virag. Our main results are as follows. (i) It was shown by B. Weiss that for an arbitrary countable infinite group G, any free ergodic probability measure preserving G-system admits a minimal model. In contrast we show here, using URS's, that for the lamplighter group there is an ergodic measure preserving action which does not admit a minimal model. (ii) For an arbitrary countable group G, every URS can be realized as the stability system of some topologically transitive G-system.Comment: To appear in the AMS Proceedings of the Conference on Recent Trends in Ergodic Theory and Dynamical System

Highly Transitive Actions of Out(Fn)

An action of a group on a set is called k-transitive if it is transitive on ordered k-tuples and highly transitive if it is k-transitive for every k. We show that for n>3 the group Out(Fn) = Aut(Fn)/Inn(Fn) admits a faithful highly transitive action on a countable set.Comment: 14 pages, (minor corrections to match final published version

An Aschbacher--O'Nan--Scott theorem for countable linear groups

The purpose of this note is to extend the classical Aschbacher--O'Nan--Scott theorem for finite groups to the class of countable linear groups. This relies on the analysis of primitive actions carried out in a previous paper. Unlike the situation for finite groups, we show here that the number of primitive actions depends on the type: linear groups of almost simple type admit infinitely (and in fact unaccountably) many primitive actions, while affine and diagonal groups admit only one. The abundance of primitive permutation representations is particularly interesting for rigid groups such as simple and arithmetic ones.Comment: 8 page