Number of ergodicity phase transitions in the uniform-noise perturbation of Gács’s cellular automaton

Determine the number of ergodicity phase transitions exhibited by the probabilistic cellular automaton obtained by perturbing Gács’s one-dimensional cellular automaton with uniform noise of rate ε across ε∈[0,1].

Background

Gács’s one-dimensional cellular automaton provides a counterexample to the positive rates conjecture by remaining non-ergodic under sufficiently small noise. It is known that for sufficiently large noise any perturbed cellular automaton is ergodic, but the detailed structure of phase transitions for Gács’s CA as ε varies is not established.

The present paper constructs a different cellular automaton T that demonstrably has at least two phase transitions, highlighting that multiple transitions are possible, but it leaves open how many transitions occur for the classical Gács example itself.