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

M. Courbage; B. Kamiński

Studia Mathematica, Tome 151 (2002), p. 285-295 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Publié le : 2002-01-01

Zbl 1016.37006

EUDML-ID : urn:eudml:doc:284837

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     title = {On the directional entropy of $\mathbb{Z}$$^2$-actions generated by cellular automata},
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     year = {2002},
     pages = {285-295},
     zbl = {1016.37006},
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M. Courbage; B. Kamiński. On the directional entropy of ℤ²-actions generated by cellular automata. Studia Mathematica, Tome 151 (2002) pp. 285-295. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm153-3-5/