Automorphisms of -free and other Toeplitz shifts

Abstract

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].