Relative generators for the action of a countable abelian group on a Lebesgue space
Generators of perfect σ-algebras of -actions
An axiomatic definition of the entropy of a -action on a Lebesgue space
On regular generator of Z²-actions in exhaustive partitions
A representation theorem for perfect partitions of -actions with finite entropy
Some properties of coalescent automorphisms of a Lebesgue space
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On abelian groups of transformations with completely positive entropy
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Spectrum of multidimensional dynamical systems with positive entropy
Applying methods of harmonic analysis we give a simple proof of the multidimensional version of the Rokhlin-Sinaǐ theorem which states that a Kolmogorov -action on a Lebesgue space has a countable Lebesgue spectrum. At the same time we extend this theorem to -actions. Next, using its relative version, we extend to -actions some other general results connecting spectrum and entropy.
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.
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 extremal and perfect σ-algebras for flows
It is shown that there exists a flow on a Lebesgue space with finite entropy and an extremal σ-algebra of it which is not perfect.
Regular generators for multidimensional dynamical systems
On the directional entropy of ℤ²-actions generated by cellular automata
We show that for any cellular automaton (CA) ℤ²-action Φ on the space of all doubly infinite sequences with values in a finite set A, determined by an automaton rule , l,r ∈ ℤ, l ≤ r, and any Φ-invariant Borel probability measure, the directional entropy , v⃗= (x,y) ∈ ℝ², is bounded above by if and by in the opposite case, where , . We also show that in the class of permutative CA-actions the bounds are attained if the measure considered is uniform Bernoulli.
Classification des automorphismes ergodiques et finitaires
On extremal and perfect σ-algebras for -actions on a Lebesgue space
We show that for every positive integer d there exists a -action and an extremal σ-algebra of it which is not perfect.
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