Cellular Automata and Groups - Errata, Additions, and Updates

T. Ceccherini-Silberstein, M. Coornaert, "Cellular automata and groups", Second edition, Springer Monographs in Mathematics, Springer, Cham, 2024, XXI + 556 pp., Hardcover ISBN 978-3-031-43327-6 Published: 14 January 2024, Softcover ISBN 978-3-031-43330-6 Due: 28 January 2025, eBook ISBN 978-3-031-43328-3 Published: 13 January 2024.

• page 244, in the proof of Lemma 6.12.3: replace "$I - M^{(2)}_{S \cup \{1_G\}} = (1 - \alpha)(I - M^{(2)}_S)$" by "$I - M^{(2)}_{S \cup \{1_G\}} = \alpha(I - M^{(2)}_S)$".
• page 248, in the proof of Lemma 6.12.8: "we have $(A_i \setminus A_ig^{-1}) \cap (A_j \setminus A_jg^{-1}) =\varnothing$" by "we have $(A_i \setminus A_ig^{-1}) \cap (A_jg^{-1} \setminus A_j) = \varnothing$".
• page 249, in the proof of Theorem 6.12.9: replace \[ \begin{split} \|(I- M_S^{(2)})x\|_2 & = \|x - \frac{1}{|S|} \sum_{s \in S} T_s^{(2)} x\|_2\\ & = \frac{1}{|S|} \|\sum_{s \in S}(x - T_s^{(2)} x)\|_2\\ & \leq \frac{1}{|S|} \sum_{s \in S}\|x - T_s^{(2)} x\|_2\\ & = \frac{1}{|S| \cdot \sqrt{|F|}} \sum_{s \in S}\|\chi_{F} - T_s^{(2)} \chi_{F}\|_2\\ \text{(by Lemma 6.12.6(iii))} \ & = \frac{1}{|S| \cdot \sqrt{|F|}} \sum_{s \in S} 2 |F \setminus F s|^{\frac{1}{2}}\\ & = \frac{1}{|S|} \sum_{s \in S} 2 \left(\frac{|F \setminus F s|}{|F|}\right)^{\frac{1}{2}}\\ & = \frac{2}{|F|^{\frac{1}{2}}} \varepsilon^{\frac{1}{2}}\\ & < \varepsilon. \end{split} \] by \[ \begin{split} \|(I- M_S^{(2)})x\|_2 & = \|x - \frac{1}{|S|} \sum_{s \in S} T_s^{(2)} x\|_2\\ & = \frac{1}{|S|} \|\sum_{s \in S}(x - T_s^{(2)} x)\|_2\\ & \leq \frac{1}{|S|} \sum_{s \in S}\|x - T_s^{(2)} x\|_2\\ & = \frac{1}{|S| \cdot \sqrt{|F|}} \sum_{s \in S}\|\chi_{F} - T_s^{(2)} \chi_{F}\|_2\\ \text{(by Lemma 6.12.6(iii))} \ & = \frac{1}{|S| \cdot \sqrt{|F|}} \sum_{s \in S} \left(2|F \setminus F s|\right)^{\frac{1}{2}}\\ & = \frac{1}{|S|} \sum_{s \in S} \left(\frac{2|F \setminus F s|}{|F|}\right)^{\frac{1}{2}}\\ & < \varepsilon. \end{split} \]
• page 251, in the proof of Theorem 6.12.9: replace "Thus, by virtue of the implication (e) $\Rightarrow$ (c) in Proposition 6.12.2" by "Thus, by virtue of the implication (e) $\Rightarrow$ (d) in Proposition 6.12.2".
• page 250, in the proof of Theorem 6.12.9: replace \[ \mu(\Omega_s) < \frac{1}{\vert S \vert} \quad \text{ for all } s \in S, \] by \[ \mu(\Omega_s) \leq \frac{1}{2\vert S \vert} < \frac{1}{\vert S \vert} \quad \text{ for all } s \in S, \]
• page 533, reference update:
  [CCL2] T. Ceccherini-Silberstein, M. Coornaert, and H. Li, Expansive actions with specification of sofic groups, strong topological Markov property, and surjunctivity, J. Funct. Anal. 286 (2024), no. 9, Paper No. 110376, 26 pp.
• page 533, reference update:
  [CCP4] T. Ceccherini-Silberstein, M. Coornaert, and X. K. Phung, Invariant sets and nilpotency of endomorphisms of algebraic sofic shifts, Ergodic Theory Dynam. Systems 44 (2024), no. 10, 2859-2900.
• page 533, reference update:
  [CCP5] T. Ceccherini-Silberstein, M. Coornaert, and X.K. Phung, First-order model theory and Kaplansky's stable finiteness conjecture, Groups, Geom. Dyn. (to appear), arXiv:2310.09451.
• page 538, reference update:
  [Phu5] X.K. Phung, Symbolic group varieties and dual surjunctivity, Groups Geom. Dyn. 18 (2024), no. 1, 213-234.
• page 538, reference update:
  [Phu10] X.K. Phung, On linear non-uniform cellular automata: duality and dynamics, Linear Algebra Appl. 688 (2024), 78-103.