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Spectrum of multidimensional dynamical systems with positive entropy

B. KamińskiP. Liardet — 1994

Studia Mathematica

Applying methods of harmonic analysis we give a simple proof of the multidimensional version of the Rokhlin-Sinaǐ theorem which states that a Kolmogorov d -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

F. BlanchardB. 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 of ℤ²-actions generated by cellular automata

M. CourbageB. 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 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.

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