Growth of jointly periodic points I

Definitions and mathematical statement

For $N\ge2$ let $X_N=\{0,\ldots,N-1\}^{\mathbb Z}$ and $(\sigma_Nx)_j=x_{j+1}$. Let $F:X_N\to X_N$ be a surjective cellular automaton, that is, a continuous map commuting with $\sigma_N$, or equivalently a map specified by a finite local rule. For each $k\ge1$ put \[ P_k(F)=\#\{x\in X_N:\sigma_N^k x=x \text{ and }F^m x=x\text{ for some }m\ge1\}, \qquad \nu(F)=\limsup_{k\to\infty}P_k(F)^{1/k}. \] The temporal period $m$ may depend on $x$ and is not bounded in this definition. Spatial periods dividing $k$ are counted, not just configurations of least spatial period $k$.

Must every such onto automaton satisfy $\nu(F)>1$? Equivalently, for each $F$ must there be some $c>1$ for which $P_k(F)\ge c^k$ for infinitely many $k$? The constant may depend on the alphabet and the local rule.

The finite set $\operatorname{Fix}(\sigma_N^k)$ contains $N^k$ points and $F$ acts on it as a self-map. Here $P_k$ counts precisely the points on its directed cycles; eventual cycle entrants are excluded. Consequently $1\le\nu(F)\le N$. The problem seeks a strictly positive exponential rate, rather than merely infinitely many jointly periodic configurations.

Short English statement

Do the cycle points of an onto cellular automaton on spatially periodic configurations always grow at a strictly exponential rate?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4600043 (AMR-045-0043), 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 25.3. Original source indexed by AMR-045-0043. https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf.