On semigroups of -endomorphisms
On strong mixing and weak convergence for groups of operators
On Subadditive Processes on Direct Product of Countable Amenable Groups
On the Cohomology of a Hyperfine Action.
On the decomposition of continuous flows on the Polish space
On the directional entropy for ℤ²-actions on a Lebesgue space
We define the concept of directional entropy for arbitrary -actions on a Lebesgue space, we examine its basic properties and consider its behaviour in the class of product actions and rigid actions.
On the existence of cohomologous continuous cocycles for cocycles with values in some Lie groups
Pointwise theorems for amenable groups.
Product -actions on a Lebesgue space and their applications
We define a class of -actions, d ≥ 2, called product -actions. For every such action we find a connection between its spectrum and the spectra of automorphisms generating this action. We prove that for any subset A of the positive integers such that 1 ∈ A there exists a weakly mixing -action, d≥2, having A as the set of essential values of its multiplicity function. We also apply this class to construct an ergodic -action with Lebesgue component of multiplicity , where k is an arbitrary positive...
Random ergodic theorems with universally representative sequences
Random orderings and unique ergodicity of automorphism groups
We show that the only random orderings of finite graphs that are invariant under isomorphism and induced subgraph are the uniform random orderings. We show how this implies the unique ergodicity of the automorphism group of the random graph. We give similar theorems for other structures, including, for example, metric spaces. These give the first examples of uniquely ergodic groups, other than compact groups and extremely amenable groups, after Glasner andWeiss’s example of the group of all permutations...
Regular generators for multidimensional dynamical systems
Relative generators for the action of a countable abelian group on a Lebesgue space
Relatively perfect σ-algebras for flows
We show that for every ergodic flow, given any factor σ-algebra ℱ, there exists a σ-algebra which is relatively perfect with respect to ℱ. Using this result and Ornstein's isomorphism theorem for flows, we give a functorial definition of the entropy of flows.
Representations of flows and generalized Rudolph's theorem [Abstract of thesis]
Rigidity of Furstenberg entropy for semisimple Lie group actions
Rigidity of some translations on homogeneous spaces.
Rigidity results for Bernoulli actions and their von Neumann algebras
Using very original methods from operator algebras, Sorin Popa has shown that the orbit structure of the Bernoulli action of a property (T) group, completely remembers the group and the action. This information is even essentially contained in the crossed product von Neumann algebra. This is the first von Neumann strong rigidity theorem in the literature. The same methods allow Popa to obtain II factors with prescribed countable fundamental group.
Simple systems are disjoint from Gaussian systems
We prove the theorem promised in the title. Gaussians can be distinguished from simple maps by their property of divisibility. Roughly speaking, a system is divisible if it has a rich supply of direct product splittings. Gaussians are divisible and weakly mixing simple maps have no splittings at all so they cannot be isomorphic. The proof that they are disjoint consists of an elaboration of this idea, which involves, among other things, the notion of virtual divisibility, which is, more or less,...
Some remarks on the metrical transitivity of invariant and quasi-invariant measures