Automorphisms of automatic shifts

Abstract

In this article we continue the study of automorphism groups of constant length substitution shifts and also their topological factors. We show that up to conjugacy, all roots of the identity map are letter exchanging maps, and all other nontrivial automorphisms arise from twisted compressions of another constant length substitution. We characterise the group of roots of the identity in both the measurable and topological setting. Finally, we show that any topological factor of a constant length substitution shift is topologically conjugate to a constant length substitution shift via a letter-to-letter code.1991 Mathematics Subject Classification. Primary 37B10, 11B85, Secondary 37B05, 37B15.

Automaticity of uniformly recurrent substitutive sequences

“…The eigenvalues of the incidence matrix M ψ are given by 4,1,0,0, and M ψ has two simple Jordan blocks corresponding to the eigenvalue 0, so s = 1. Since ψ is primitive and left-proper, and (16,16,21,11) = (16,16,20,12…”

“…Recall that a complex number λ is a (topological) dynamical eigenvalue of a subshift X if there exists a continuous function f : X → Z such that f • T = λf , where T denotes the shift map. Thanks to earlier work of Dekking [7] and recent work of the second author and Yassawi [21], the dynamical eigenvalues of minimal automatic systems are well understood. For an infinite minimal k-automatic system X its eigenvalues are given by k n -th roots of unity, n 1 and h-th roots of unity, where h is an integer coprime with k known as the height of X.…”

“…For an infinite minimal k-automatic system X its eigenvalues are given by k n -th roots of unity, n 1 and h-th roots of unity, where h is an integer coprime with k known as the height of X. This is the same as saying that the additive group Z k × Z/hZ is the maximal equicontinuous factor of X, where Z k is the ring of k-adic integers (we refer to [21] for more details). One may often show that a given (minimal) substitutive system is not automatic by computing its (dynamical) eigenvalues as described e.g.…”

“…We will also need the following recent result of the second author and Yassawi, which shows that dynamically two-sided minimal automatic systems and minimal purely automatic systems are the same [21]. 2 This is not true on the level of sequences: there are automatic sequences, which are not purely automatic, the most famous example being perhaps the Golay-Shapiro sequence (known also as the Rudin-Shapiro sequence) [3,Ex.…”

“…The proof in[21] uses the invertibility of the automatic system, so it is not immediate to transfer Theorem 2.5 to the one-sided case.…”