On the garden of Eden theorem for \(\mathscr{B}\)-free subshifts (Q2677815)

This paper relates dynamical and number-theoretical properties of subshifts. Here a \textit{subshift} is a closed subspace of \(\{0,1\}^{\mathbb Z}\) that is invariant under the shift \((a_i)_i\mapsto(a_{i+1})_i\). For a subset \(\mathscr B\subseteq\mathbb N\), the associated \textit{\(\mathscr B\)-free subshift} is constructed as follows: define \(\eta\colon\mathbb Z\to\{0,1\}\) by \[ \eta(n)=\begin{cases} 1 & \text{ if }n\ge1\text{ and is not divisible by any element of }\mathscr B,\\ 0 & \text{ otherwise},\end{cases} \] and let \(X_\eta\) be the smallest closed, shift-invariant subset of \(\{0,1\}^{\mathbb Z}\) that contains \(\eta\). The present paper focuses on \textit{cellular automata}, namely shift-commuting continuous self-maps of the shift. The celebrated ``Garden of Eden'' theorem by \textit{E. F. Moore} [in: Proc. Sympos. Appl. Math. 14, 17--33 (1962; Zbl 0126.32408)] and \textit{J. Myhill} [Proc. Am. Math. Soc. 14, 685--686 (1963; Zbl 0126.32501)] asserts that a cellular automaton is surjective if and only if it is \textit{pre-injective}, namely it takes different values on configurations that differ on a finite, non-empty set. Originally claimed for the full shift \(\{0,1\}^{\mathbb Z}\), it was shown to hold in a much larger generality, for example for strongly irreducible subshifts of finite type. The main contribution of the authors (Theorem~1.1) is to show that the ``Garden of Eden'' theorem holds for \(X_\eta\) whenever \(\mathscr B\) is an \textit{Erdős} set, namely when \(\mathscr B\) is infinite and \(\sum_{b\in\mathscr B}1/b\) converges. Furthermore, they give strong structural results on the associated cellular automata: the only surjective ones are powers of the shift, and (up to powers of the shift) every cellular automaton is monotone (it decreases coordinates pointwise).