Goles, E., MontalvaMedel, M., MacLean, S., & Mortveit, H. (2018). Block Invariance in a Family of Elementary Cellular Automata. J. Cell. Autom., 13(12), 15–32.
Abstract: We study the steady state invariance of elementary cellular automata (ECA) under different deterministic updating schemes. Specifically, we study a family of eleven ECA whose steady state invariance were left under conjecture in [2].

Goles, E., MontalvaMedel, M., Mortveit, H., & RamirezFlandes, S. (2015). Block Invariance in Elementary Cellular Automata. J. Cell. Autom., 10(12), 119–135.
Abstract: Consider an elementary cellular automaton (ECA) under periodic boundary conditions. Given an arbitrary partition of the set of vertices we consider the block updating, i.e. the automaton's local function is applied from the first to the last set of the partition such that vertices belonging to the same set are updated synchronously. The automaton is said blockinvariant if the set of periodic configurations is independent of the choice of the block updating. When the sets of the partition are singletons we have the sequential updating: vertices are updated one by one following a permutation pi. In [5] the authors analyzed the piinvariance of the 2(8) = 256 possible ECA rules (or the 88 nonredundant rules subset). Their main result was that for all n > 3, exactly 41 of these nonredundant rules are piinvariant. In this paper we determine the subset of these 41 rules that are block invariant. More precisely, for all n > 3, exactly 15 of these rules are block invariant. Moreover, we deduce that block invariance also implies that the attractor structure itself is independent of the choice of the block update.

MacLean, S., MontalvaMedel, M., & Goles, E. (2019). Block invariance and reversibility of one dimensional linear cellular automata. Adv. Appl. Math., 105, 83–101.
Abstract: Consider a onedimensional, binary cellular automaton f (the CA rule), where its n nodes are updated according to a deterministic block update (blocks that group all the nodes and such that its order is given by the order of the blocks from left to right and nodes inside a block are updated synchronously). A CA rule is block invariant over a family F of block updates if its set of periodic points does not change, whatever the block update of F is considered. In this work, we study the block invariance of linear CA rules by means of the property of reversibility of the automaton because such a property implies that every configuration has a unique predecessor, so, it is periodic. Specifically, we extend the study of reversibility done for the Wolfram elementary CA rules 90 and 150 as well as, we analyze the reversibility of linear rules with neighbourhood radius 2 by using matrix algebra techniques. (C) 2019 Elsevier Inc. All rights reserved.
