  We study non-equilibrium defect accumulation dynamics on a cellular automaton
trajectory: a branching walk process in which a defect creates a successor on
any neighborhood site whose update it affects. On an infinite lattice, defects
accumulate at different exponential rates in different directions, giving rise
to the Lyapunov profile. This profile quantifies instability of a cellular
automaton evolution and is connected to the theory of large deviations. We
rigorously and empirically study Lyapunov profiles generated from random
initial states. We also introduce explicit and computationally feasible
variational methods to compute the Lyapunov profiles for periodic
configurations, thus developing an analogue of Floquet theory for cellular
automata.
