Contractible subshifts

Abstract

We introduce the notion of a contractible subshift. This is a strengthening of the notion of strong irreducibility, where we require that the gluings are given by a block map. We show that a subshift is a retract of a full shift if and only if it is a contractible SFT with a fixed point. For many groups, including virtually polycyclic groups, metabelian Baumslag-Solitar groups and the lamplighter group, contractibility implies dense periodic points. We introduce a ``homotopy theory'' framework for working with this notion, and ``contractibility'' is in fact simply an analog of the usual contractibility in algebraic topology. We also explore the symbolic dynamical analogs of homotopy equivalence and strong contractibility of subshifts (corresponding to the notion of equiconnectedness in topology). Contractibility is implied by the map extension property of Meyerovitch, and among SFTs, it implies the finite extension property of Brice\~no, McGoff and Pavlov. We include thorough comparisons with these classes. We also encounter some new geometric notions, in particular a periodic variant of Gromov's asymptotic dimension of a group.