Exercises in Cellular Automata and Groups - Errata, Additions, and Updates

T. Ceccherini-Silberstein, M. Coornaert, "Exercises in Cellular automata and groups", Springer Monographs in Mathematics, Springer, Cham, 2023, XVII + 627 pp., eBook ISBN 978-3-031-10391-9, DOI 10.1007/978-3-031-10391-9, Hardcover ISBN 978-3-031-10390-2, Softcover ISBN 978-3-031-10393-3.

• page 69, Exercise 1.76: add the following question
  " (e) Show that $X$ is strongly irreducible if and only if the following condition holds: there exists an integer $n_0 \geq 0$ such that, for all words $u$ and $v$ in $L(X)$, there exists a word $w \in A^*$ of length $\ell(w) = n_0$ such that $u w v \in L(X)$. "
• page 70, exercise 1.76: add the following solution
  (e) The condition is necessary by (d). Conversely, suppose that there exists an integer $n_0 \geq 0$ satisfying the condition stated in (e). Let $u,v \in L(X)$ and let $n \geq n_0$ be an integer. By Exercise~1.74.(e), there exists a word $u' \in A^*$ of length $n - n_0$ such that $uu' \in L(X)$. By our hypothesis, there exists a word $w' \in A^*$ of length $n_0$ such that $uu'w'v \in L(X)$. Then the word $w := u'w'$ has length $n$ and satisfies $uwv = uu'w'v \in L(X)$. By (d), this shows that $X$ is strongly irreducible. "
• page 289, lines 12-15: replace " \[ T_j \subset \eta_{F_j \Delta}(Y) \times \pi_{F_j\Delta^2 \setminus F_j}(X) \subset \eta_{F_j }(Y) \times A^{F_j\Delta \setminus F_j} \times A^{F_j\Delta^2 \setminus F_j}, \] so that \[ \log |T_j| \leq \log |\eta_{F_j}(Y) \times A^{F_j\Delta \setminus F_j} \times A^{F_j\Delta^2 \setminus F_j}| = \log|\eta_{F_j }(Y)| + (|F_j\Delta \setminus F_j| + |F_j\Delta^2 \setminus F_j|) \log|A| \] for all $j \in J$." by " \[ T_j \subset \eta_{F_j \Delta}(Y) \times \pi_{F_j\Delta^2 \setminus F_j}(X) \subset \eta_{F_j }(Y) \times B^{F_j\Delta \setminus F_j} \times A^{F_j\Delta^2 \setminus F_j}, \] so that \[ \log |T_j| \leq \log |\eta_{F_j}(Y) \times B^{F_j\Delta \setminus F_j} \times A^{F_j\Delta^2 \setminus F_j}| = \log|\eta_{F_j }(Y)| + |F_j\Delta \setminus F_j| \log|B| + |F_j\Delta^2 \setminus F_j| \log|A| \] for all $j \in J$. "
• page 289, line -1: replace " \[ \frac{\log |T_{j_0}|}{|F_{j_0}|} < \frac{\log |Z_{j_0}|}{F_{j_0}|} \] " by " \[ \frac{\log |T_{j_0}|}{|F_{j_0}|} < \frac{\log |Z_{j_0}|}{|F_{j_0}|} \] "
• page 514, line 4: replace "$a^n = 1_R$" by "$a^n = 0_R$".

Last Update: November 9, 2026