Symbol Frequencies in Surjective Cellular Automata

: Hellouin de Menibus, Benjamin; Salo, Ville; Törmä, Ilkka

: Riva, Sara; Richard, Adrien

: International Workshop on Cellular Automata and Discrete Complex Systems

Publisher: Springer Nature Switzerland

: 2025

Lecture Notes in Computer Science

: Cellular Automata and Discrete Complex Systems: 31st IFIP WG 1.5 International Workshop, AUTOMATA 2025, Lille, France, June 30 – July 2, 2025, Proceedings

: 15831

: 154

: 170

: 978-3-032-01569-3

: 978-3-032-01570-9

: 0302-9743

: 1611-3349

DOI: https://doi.org/10.1007/978-3-032-01570-9_11

: https://doi.org/10.1007/978-3-032-01570-9_11

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.