On the directional entropy of ${\Bbb Z}^2$-actions generated by cellular automata

M. Courbage¹ ; B. Kamiński²

¹ Laboratoire de Physique Théorique de la Matière Condensée Université Paris VII Tour 24, 5ème étage, porte 506 B.P. 7020 Place Jussieu 75251 Paris Cedex, France
² Faculty of Mathematics and Computer Science Nicholas Copernicus University Chopina 12/18 87-100 Toruń, Poland

Studia Mathematica, Tome 153 (2002) no. 3, pp. 285-295

Résumé

We show that for any cellular automaton (CA) ${{\mathbb Z}}^{2}$-action ${\mit \Phi }$ on the space of all doubly infinite sequences with values in a finite set $A$, determined by an automaton rule $F=F_{[l,r]}$, $l,r\in {{\mathbb Z}}$, $l\leq r$, and any ${\mit \Phi }$-invariant Borel probability measure, the directional entropy $h_{\vec {v}}({\mit \Phi }), \vec {v}=(x,y) \in {{\mathbb R}}^{2}$, is bounded above by ${\mathop {\rm max}}(|z_{l}|,|z_{r}|)\mathop {\rm log}\nolimits \#A$ if $z_{l}z_{r}\geq 0$ 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.

Détail

DOI : 10.4064/sm153-3-5

Keywords: cellular automaton mathbb action mit phi space doubly infinite sequences values finite set determined automaton rule mathbb leq mit phi invariant borel probability measure directional entropy vec mit phi vec mathbb bounded above mathop max mathop log nolimits geq z opposite where class permutative ca actions bounds attained measure considered uniform bernoulli