Symbol Frequencies in Surjective Cellular Automata

TL;DR

We address how surjective cellular automata $F$ act on shift-invariant measures and whether iterates $F^t\mu$ preserve full support in the weak-* limit for product measures with full support. The study introduces a preimage-frequency tool and a $C^A_B(F,r,m)$ (the $(A,B,r)$-correlation) to quantify how many $A$-symbol occurrences appear in preimages of $B$-symbols, leveraging entropy conservation $h(F\mu)=h(\mu)$ and $F\lambda=\lambda$. The main contributions include constructing a sequence of reversible binary CAs $(F_n)$ with $F_n([w])\to 0$ for some word $w$, showing that no full-support weak-* limit can exist in that setting, and producing an ergodic full-support measure $\mu$ for the two-neighbor XOR CA with $F^t\mu\to\delta_0$ (and a contrasting limit behavior in another instance). The work also shows that the correlation framework yields partial domination results (e.g., identity CA maximizing $(A,A)$-correlation) and that these tools do not fully resolve whether a full-support limit point must exist, ultimately clarifying limitations of entropy-based intuitions and providing a new analytic toolkit for analyzing limit measures under surjective CAs.

Abstract

We study the behavior of probability measures under iteration of a surjective cellular automaton. We solve the following question in the negative: if the initial measure is ergodic and has full support, do all weak-* limit points of the sequence of measures have full support as well? The initial measure of our solution is not a product measure, and in this case the question remains open. To this end, we present a tool for studying the frequencies of symbols in preimages of surjective cellular automata, and prove some basic results about it. However, we show that by itself it is not enough to solve the stricter question in the positive.