On a zero speed sensitive cellular automaton

TL;DR

This paper demonstrates the existence of a sensitive cellular automaton with perturbations that propagate at asymptotically zero speed for almost all configurations, revealing new insights into automaton sensitivity and entropy.

Contribution

It introduces a natural invariant measure showing that certain sensitive automata can have zero-speed perturbation propagation, contrasting with positively expansive automata.

Findings

Lyapunov Exponents are zero for almost all configurations.

The automaton's measurable entropy is zero.

Perturbations propagate at positive speed only in positively expansive automata.

Abstract

Using an unusual, yet natural invariant measure we show that there exists a sensitive cellular automaton whose perturbations propagate at asymptotically null speed for almost all configurations. More specifically, we prove that Lyapunov Exponents measuring pointwise or average linear speeds of the faster perturbations are equal to zero. We show that this implies the nullity of the measurable entropy. The measure m we consider gives the m-expansiveness property to the automaton. It is constructed with respect to a factor dynamical system based on simple "counter dynamics". As a counterpart, we prove that in the case of positively expansive automata, the perturbations move at positive linear speed over all the configurations.