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

• Autorzy: M. Courbage , B. Kamiński
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 37A15: General groups of measure-preserving transformations \SeeMainly{}
• 28D20: Entropy and other invariants
• 37A35: Entropy and other invariants, isomorphism, classification

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2002
• Tom: 153
• Numer: 3
• Strony: 285-295

Daty

wydano

2002

Twórcy

autor

M. Courbage

• 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

autor

B. Kamiński

• Faculty of Mathematics and Computer Science, Nicholas Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

Identyfikatory

DOI

10.4064/sm153-3-5