Entropy of Gaussian actions for countable Abelian groups
Ergodic invariant measures for actions of SL(2, Z)
Ergodic Properties of Certain Surjective Cellular Automata.
Ergodic theory, semisimple Lie groups and foliations by manifolds of negative curvature
Estimates of capacity of self-similar measures
We give lower and upper estimates of the capacity of self-similar measures generated by iterated function systems where are bi-lipschitzean transformations.
Extensions of Cocycles for Hyperfinite Actions and Applications.
Extensions of probability-preserving systems by measurably-varying homogeneous spaces and applications
We study a generalized notion of a homogeneous skew-product extension of a probability-preserving system in which the homogeneous space fibres are allowed to vary over the ergodic decomposition of the base. The construction of such extensions rests on a simple notion of 'direct integral' for a 'measurable family' of homogeneous spaces, which has a number of precedents in older literature. The main contribution of the present paper is the systematic development of a formalism for handling such extensions,...
Finitary orbit equivalence and measured Bratteli diagrams
We prove a strengthened version of Dye's theorem on orbit equivalence, showing that if the transformation structures are represented as finite coordinate change equivalence relations of ergodic measured Bratteli diagrams, then there is a finitary orbit equivalence between these diagrams.
Finite rank actions and the loosely Bernoulli property.
Finiteness of Ergodic Unitarily Invariant Measures on Spaces of Infinite Matrices
The main result of this note, Theorem 1.3, is the following: a Borel measure on the space of infinite Hermitian matrices, that is invariant and ergodic under the action of the infinite unitary group and that admits well-defined projections onto the quotient space of “corners" of finite size, must be finite. A similar result, Theorem 1.1, is also established for unitarily invariant measures on the space of all infinite complex matrices. These results imply that the infinite Hua-Pickrell measures...
Generalized Calderón conditions and regular orbit spaces
The construction of generalized continuous wavelet transforms on locally compact abelian groups A from quasi-regular representations of a semidirect product group G = A ⋊ H acting on L²(A) requires the existence of a square-integrable function whose Plancherel transform satisfies a Calderón-type resolution of the identity. The question then arises under what conditions such square-integrable functions exist. The existing literature on this subject leaves a gap between sufficient and necessary criteria....
Generators for Amenable Group Actions.
Generators of an Abelian Group of Invertible Measure-Preserving Transformations.
Generators of perfect σ-algebras of -actions
Groups of Measure Preserving Transformations.
Groups of Measure Preserving Transformations. II.
Homomorphisms of ergodic group actions and conjugacy of skew product actions.
Hopf's theorem on invariant measures for a group of transformations
Identification of periodic and cyclic fractional stable motions
We consider an important subclass of self-similar, non-gaussian stable processes with stationary increments known as self-similar stable mixed moving averages. As previously shown by the authors, following the seminal approach of Jan Rosiński, these processes can be related to nonsingular flows through their minimal representations. Different types of flows give rise to different classes of self-similar mixed moving averages, and to corresponding general decompositions of these processes. Self-similar...
Infinite ergodic index -actions in infinite measure
We construct infinite measure preserving and nonsingular rank one -actions. The first example is ergodic infinite measure preserving but with nonergodic, infinite conservative index, basis transformations; in this case we exhibit sets of increasing finite and infinite measure which are properly exhaustive and weakly wandering. The next examples are staircase rank one infinite measure preserving -actions; for these we show that the individual basis transformations have conservative ergodic Cartesian...