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.
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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.
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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 $ℤ^∞$-actions. Next, using its relative version, we extend to $ℤ^∞$-actions some other general results connecting spectrum and entropy.
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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 $h_{v⃗}(Φ)$, v⃗= (x,y) ∈ ℝ², is bounded above by $max(|z_{l}|,|z_{r}|) log #A$ if $z_{l}z_{r} ≥ 0$ and by $|z_{r} - z_{l}|$ in the opposite case, where $z_{l} = x + ly$, $z_{r} = x + ry$. 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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