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

M. Courbage, and B. Kamiński. "On the directional entropy of ℤ²-actions generated by cellular automata." Studia Mathematica 153.3 (2002): 285-295. <http://eudml.org/doc/284837>.

@article{M2002,
abstract = {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.},
author = {M. Courbage, B. Kamiński},
journal = {Studia Mathematica},
keywords = {cellular automata; directional entropy; invariant measure},
language = {eng},
number = {3},
pages = {285-295},
title = {On the directional entropy of ℤ²-actions generated by cellular automata},
url = {http://eudml.org/doc/284837},
volume = {153},
year = {2002},
}

TY - JOUR
AU - M. Courbage
AU - B. Kamiński
TI - On the directional entropy of ℤ²-actions generated by cellular automata
JO - Studia Mathematica
PY - 2002
VL - 153
IS - 3
SP - 285
EP - 295
AB - 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.
LA - eng
KW - cellular automata; directional entropy; invariant measure
UR - http://eudml.org/doc/284837
ER -