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

Volume 153 / 2002

M. Courbage, B. Kamiński Studia Mathematica 153 (2002), 285-295 MSC: Primary 28D20; Secondary 37A15, 37A35. DOI: 10.4064/sm153-3-5


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.


  • M. CourbageLaboratoire 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
  • B. KamińskiFaculty of Mathematics and Computer Science
    Nicholas Copernicus University
    Chopina 12/18
    87-100 Toruń, Poland

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