Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

A cellular automaton is monotonous if it preserves the order on configurations inherited from an order on

states. If at first sight, the significantly sparse aspect of that product order lead us to believe that mono-

tonicity would not be that restrictive, the fact that cellular automata are characterised by a local rule

applied shift-invariantly makes the impact of this constraint more remarkable than expected. After intro-

ducing the basic concepts of our work, we first state general results about monotonous cellular automata

on Z with especially the fact that for any shift-ergodic measure of full support, the sequence converges in

Cesàro mean. Then, taking inspiration from previous constructions to show some (/un)computability re-

sults of the literature, and "monotonising" them, we get some upper bound on those "good behaviours".

Finally, we consider some toy example of majority cellular automata, on an asymmetrical neighbourhood,

to try to find a simple example to illustrate a result of ours linking directional convergence of all orbits

with convergence in measure.