Exact number of phase transitions for the constructed automaton T

Ascertain whether the cellular automaton T constructed in the paper exhibits exactly two ergodicity phase transitions under perturbation by uniform noise as ε varies, or whether additional phase transitions occur.

Background

The paper constructs a cellular automaton T which, under uniform-noise perturbation, is ergodic for sufficiently small ε, non-ergodic at a certain intermediate ε, and ergodic again for sufficiently large ε, thereby proving at least two phase transitions. However, the authors do not establish that these are the only transitions that occur for T.

Clarifying the exact number of phase transitions for T would sharpen the main result by fully characterizing the ergodicity behavior of T across all noise rates.