Subshifts with sparse traces

We study two-dimensional subshifts whose horizontal trace (a.k.a. projective subdynamics) contains only points of finite support. Our main result is a classification result for such subshifts satisfying a minimality property. As corollaries, we obtain new proofs for various known results on traces of SFTs, nilpotency and decidability of cellular automata, topological full groups and the subshift of prime numbers. We also construct various (sofic) examples illustrating the concepts.Comment: 43 pages, 9 figures. Fixed some broken definitions, elaborated on some proofs (kept the informal style). Added an example on path extraction, results otherwise unchanged. Comments welcome

A note on directional closing

We show that directional closing in the sense of Guillon-Kari-Zinoviadis and Franks-Kra is not closed under conjugacy. This implies that being polygonal in the sense of Franks-Kra is not closed under conjugacy.Comment: 4 pages, 2 figure

Minimal subshifts with a language pivot property

We construct a binary minimal subshift whose words of length n form a connected subset of the Hamming graph for each n.Comment: 6 page

Toeplitz subshift whose automorphism group is not finitely generated

We compute an explicit representation of the (topological) automorphism group or a particular Toeplitz subshift. The automorphism group is a (non-finitely generated) subgroup of rational numbers under addition and the shift map corresponds to the rational number 1. The group is the additive subgroup of the rational numbers generated by the powers of 5/2.Comment: 22 pages. Comments and corrections welcome

Decidability and Universality of Quasiminimal Subshifts

We introduce the quasiminimal subshifts, subshifts having only finitely many subsystems. With $\mathbb{N}$-actions, their theory essentially reduces to the theory of minimal systems, but with $\mathbb{Z}$-actions, the class is much larger. We show many examples of such subshifts, and in particular construct a universal system with only a single proper subsystem, refuting a conjecture of [Delvenne, K\r{u}rka, Blondel, '05].Comment: 40 pages, 1 figure, submitted to JCS

On Nilpotency and Asymptotic Nilpotency of Cellular Automata

We prove a conjecture of P. Guillon and G. Richard by showing that cellular automata that eventually fix all cells to a fixed symbol 0 are nilpotent on S^Z^d for all d. We also briefly discuss nilpotency on other subshifts, and show that weak nilpotency implies nilpotency in all subshifts and all dimensions, since we do not know a published reference for this.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249

When are group shifts of finite type?

It is known that a group shift on a polycyclic group is necessarily of finite type. We show that, for trivial reasons, if a group does not satisfy the maximal condition on subgroups, then it admits non-SFT abelian group shifts. In particular, we show that if group is elementarily amenable or satisfies the Tits alternative, then it is virtually polycyclic if and only if all its group shifts are of finite type. Our theorems are minor elaborations of results of Schmidt and Osin.Comment: 8 pages; fixed a typo IN THE TITL

A note on subgroups of automorphism groups of full shifts

We discuss the set of subgroups of the automorphism group of a full shift, and submonoids of its endomorphism monoid. We prove closure under direct products in the monoid case, and free products in the group case. We also show that the automorphism group of a full shift embeds in that of an uncountable sofic shift. Some undecidability results are obtained as corollaries

A Characterization of Cellular Automata Generated by Idempotents on the Full Shift

In this article, we discuss the family of cellular automata generated by so-called idempotent cellular automata (CA G such that G^2 = G) on the full shift. We prove a characterization of products of idempotent CA, and show examples of CA which are not easy to directly decompose into a product of idempotents, but which are trivially seen to satisfy the conditions of the characterization. Our proof uses ideas similar to those used in the well-known Embedding Theorem and Lower Entropy Factor Theorem in symbolic dynamics. We also consider some natural decidability questions for the class of products of idempotent CA.Comment: will be presented in CSR 201

Hard Asymptotic Sets for One-Dimensional Cellular Automata

We prove that the (language of the) asymptotic set (and the nonwandering set) of a one-dimensional cellular automaton can be \SIGMA^1_1-hard. We do not go into much detail, since the constructions are relatively standard