Sparse jointly periodic points

Definitions and mathematical statement

For an integer $N\ge2$, write $X_N=\{0,\ldots,N-1\}^{\mathbb Z}$ and let $\sigma_N$ be its left shift. A one-dimensional cellular automaton $F:X_N\to X_N$ is a continuous shift-commuting map, equivalently one given by a finite local rule. For $k\ge1$ count \[ P_k(F)=\#\{x\in X_N:\sigma_N^k x=x \text{ and }F^m x=x\text{ for some }m\ge1\}, \qquad \nu(F)=\limsup_{k\to\infty}P_k(F)^{1/k}. \] Construct a surjective $F$ for some finite $N\ge2$ such that $\nu(F)<N$, or disprove the asserted existence.

This is an existential question about both the alphabet and the rule. A successful construction must prove global surjectivity and a strict asymptotic gap: equivalently, there must be $c<N$ and $k_0$ with $P_k(F)\le c^k$ for every $k\ge k_0$. Finite computations showing small counts at selected periods do not establish the required limsup inequality.

The count includes spatial periods dividing $k$ and all temporal cycle lengths. Since $\operatorname{Fix}(\sigma_N^k)$ has $N^k$ points, the threshold $N$ represents full exponential growth. Invertible automata have every point of each such finite set on a temporal cycle; thus any proposed example must use surjectivity without invertibility. The question concerns this growth deficit, without also requiring failure of density of jointly periodic configurations.

Short English statement

Construct an onto cellular automaton whose jointly periodic configurations have exponential growth strictly below the full alphabet rate.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4600045 (AMR-045-0045), 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 25.5. Original source indexed by AMR-045-0045. https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf.