On the Garden of Eden theorem for ℬ-free subshifts

Keller, Gerhard; Lemańczyk, Mariusz; Richard, Christoph; Sell, Daniel

doi:10.1007/s11856-022-2437-9

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2023

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“…We mention briefly that Mentzen [22] proves the triviality of the automorphism group for any Erdős -free subshift, that is, when is infinite, pairwise coprime and . This is extended to taut containing an infinite pairwise coprime subset in [19, 20]. This class of -free subshifts is kind of opposite to the -free Toeplitz shifts.…”

Section: Introductionmentioning

confidence: 99%

Automorphisms of -free and other Toeplitz shifts

Dymek¹,

Kasjan²,

Keller³

2023

Ergod. Th. Dynam. Sys.

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We present sufficient conditions for the triviality of the automorphism group of regular Toeplitz subshifts and give a broad class of examples from the class of ${\mathcal B}$ -free subshifts satisfying them, extending the work of Dymek [Automorphisms of Toeplitz ${\mathcal B}$ -free systems. Bull. Pol. Acad. Sci. Math.65(2) (2017), 139–152]. Additionally, we provide an example of a ${\mathcal B}$ -free Toeplitz subshift whose automorphism group has elements of arbitrarily large finite order, answering Question 11 of S. Ferenczi et al [Sarnak’s conjecture: what’s new. Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics (Lecture Notes in Mathematics, 2213). Eds. S. Ferenczi, J. Kułaga-Przymus and M. Lemańczyk. Springer, Cham, 2018, pp. 163–235].

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