Growth of jointly periodic points II

Definitions and mathematical statement

Let $N\ge2$ and $X_N=\{0,\ldots,N-1\}^{\mathbb Z}$ with left shift $\sigma_N$. Consider any surjective continuous map $F:X_N\to X_N$ commuting with the shift. Such an $F$ is a one-dimensional cellular automaton and is determined by a finite-window local rule. Define \[ J_k(F)=\{x:\sigma_N^k x=x,\quad F^m x=x\text{ for some integer }m\ge1\}, \qquad \nu(F)=\limsup_{k\to\infty}|J_k(F)|^{1/k}. \] The question is whether the universal lower bound \[ \nu(F)\ge\sqrt N \] holds for every alphabet size and every such $F$.

The count uses configurations with spatial period dividing $k$, with arbitrary positive temporal period. Distinct configurations are counted separately even when they belong to the same shift orbit or the same temporal cycle. Thus the ambient count at spatial period $k$ is $N^k$. The limsup allows oscillation as $k$ varies; the claim does not require a bound $|J_k(F)|\ge N^{k/2}$ for every $k$.

Equivalently, for each $\varepsilon>0$ there must be infinitely many $k$ with $|J_k(F)|^{1/k}>\sqrt N-\varepsilon$. Surjectivity is assumed only for the full configuration space. The proposed square-root threshold is independent of the local rule's radius, and is stronger than asking merely that $\nu(F)>1$ for each individual automaton.

Short English statement

Is the exponential growth rate of jointly periodic configurations always at least the square root of the alphabet size?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4600044 (AMR-045-0044), 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 25.4. Original source indexed by AMR-045-0044. https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf.