Periodic points of cellular automata

Definitions and mathematical statement

Fix $d\ge1$ and a finite alphabet $A$. On $X=A^{\mathbb Z^d}$ with the product topology, let $F:X\to X$ be a surjective cellular automaton: for some finite neighborhood $D\subset\mathbb Z^d$ and rule $\varphi:A^D\to A$, $(Fx)_v=\varphi((x_{v+u})_{u\in D})$ for every $v$. A point is temporally periodic if $F^k x=x$ for some $k\ge1$. It is spatially periodic if its orbit under all lattice translations is finite, equivalently if a finite-index subgroup of $\mathbb Z^d$ fixes it. A jointly periodic point satisfies both properties.

The source asks three questions. Must the temporally periodic points be dense in $X$? Must the jointly periodic points be dense? Must the union of all minimal subsystems of $(X,F)$ be dense? A minimal subsystem is a nonempty compact set $K\subset X$ with $F(K)=K$ and no proper nonempty closed invariant subset; equivalently, every forward orbit in $K$ is dense in $K$.

Density is tested against every nonempty finite-coordinate cylinder: each finite pattern must extend to a point of the indicated type. The temporal period, spatial period subgroup, or minimal set may depend on the pattern. Surjectivity is global, on the full infinite configuration space. It is not an assumption of injectivity or of surjectivity on each finite periodic lattice separately.

Short English statement

Does every onto cellular automaton have dense periodic points, dense jointly periodic points, and dense minimal subsystems?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4600041 (AMR-045-0041), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Boyle - Open Problems in Symbolic Dynamics (2008), Problem/Question/Conjecture 25.1. Original source indexed by AMR-045-0041. https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf.