Coupling from the past for exponentially ergodic one-dimensional probabilistic cellular automata

For every exponentially ergodic one-dimensional probabilistic cellular automaton with positive rates, we construct a locally defined coupling-from-the-past flow whose coalescence time has a finite exponential moment. This construction leads to a finite-size necessary and sufficient condition for exponential ergodicity of one-dimensional cellular automata. As a corollary, we prove that every sufficiently small perturbation of an exponentially ergodic one-dimensional cellular automaton is exponentially ergodic.