EUDML

Applying methods of harmonic analysis we give a simple proof of the multidimensional version of the Rokhlin-Sinaǐ theorem which states that a Kolmogorov $ℤ^{d}$ -action on a Lebesgue space has a countable Lebesgue spectrum. At the same time we extend this theorem to $ℤ^{\infty}$ -actions. Next, using its relative version, we extend to $ℤ^{\infty}$ -actions some other general results connecting spectrum and entropy.

Relatively perfect σ-algebras for flows

F. Blanchard; B. Kamiński — 1995

Studia Mathematica

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

B. Kamiński; K. Park — 1999

Studia Mathematica

We define the concept of directional entropy for arbitrary $ℤ^{2}$ -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

B. Kamiński; Z. Kowalski — 1998

Studia Mathematica

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

B. Kamiński; M. Kobus — 1986

Colloquium Mathematicae

On the directional entropy of ℤ²-actions generated by cellular automata

M. Courbage; B. Kamiński — 2002

Studia Mathematica

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 $F = F_{[l, r]}$ , l,r ∈ ℤ, l ≤ r, and any Φ-invariant Borel probability measure, the directional entropy $h_{v ⃗} (Φ)$ , v⃗= (x,y) ∈ ℝ², is bounded above by $m a x (| z_{l} |, | z_{r} |) l o g A$ if $z_{l} z_{r} \geq 0$ and by $| z_{r} - z_{l} |$ in the opposite case, where $z_{l} = x + l y$ , $z_{r} = x + r y$ . We also show that in the class of permutative CA-actions the bounds are attained if the measure considered is uniform Bernoulli.