Two points of view to study the iterates of a random configuration by a cellular automaton

ISBN 978-5-94057-377-7International audienceWe study the dynamics of the action of cellular automata on the set of shift-invariant probability measures according two points of view. First, the robustness of the simulation of a cellular automaton on a random configuration can be viewed considering the sensitivity to initial condition in the space of shift-invariant probability measures. Secondly we consider the evolution of the quantity of information in the orbit of a random initial state

Measure rigidity for algebraic bipermutative cellular automata

International audienceLet (\az,F) be a bipermutative algebraic cellular automaton. We present conditions which force a probability measure which is invariant for the $\N\times\Z$-action of $F$ and the shift map \s to be the Haar measure on \gs, a closed shift-invariant subgroup of the Abelian compact group \az. This generalizes simultaneously results of B. Host, A. Maass and S. Mart\'{\i}nez \cite{Host-Maass-Martinez-2003} and M. Pivato \cite{Pivato-2003}. This result is applied to give conditions which also force a (F,\s)-invariant probability measure to be the uniform Bernoulli measure when $F$ is a particular invertible expansive cellular automaton on \an

Entry times in automata with simple defect dynamics

In this paper, we consider a simple cellular automaton with two particles of different speeds that annihilate on contact. Following a previous work by K\r urka et al., we study the asymptotic distribution, starting from a random configuration, of the waiting time before a particle crosses the central column after time n. Drawing a parallel between the behaviour of this automata on a random initial configuration and a certain random walk, we approximate this walk using a Brownian motion, and we obtain explicit results for a wide class of initial measures and other automata with similar dynamics.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249