Relationship between boundary conditions and ergodicity

Characterize the relationship between ergodicity under one boundary driving permutation TB and ergodicity under other boundary driving permutations TB in boundary-driven semi-infinite reversible cellular automata, identifying conditions under which ergodicity coincides or differs across boundary choices.

Background

Throughout the work, ergodicity is often assessed uniformly over all boundary permutations TB for small k, and examples are given (e.g., a k=7 construction) where ergodicity depends on the specific boundary permutation.

Understanding how and when boundary choices change ergodicity outcomes is essential for extending classification results beyond small k and for interpreting numerical searches for ergodic rules.