On the Garden of Eden theorem for B-free subshifts

Abstract

We prove that on B-free subshifts, with B satisfying the Erdös condition, all cellular automata are determined by monotone sliding block codes. In particular, this implies the validity of the Garden of Eden theorem for such systems.

Dynamics of $\mathscr{B}$-free systems generated by Behrend sets. I

“…Proximality and characterization of the Erdös case It has already been noticed in [21], that whenever (X η , S ) is proximal then…”

“…The existence of such q r,t is a consequence of Dirichlet Theorem (applied to r gcd(r,t) + t gcd(r,t) ). Let (21) C 0 = {c ∈ C \ {a} : c|q r,t , for some (r, t) ∈ T }.…”

“…Corollary 1.1). Building on some recent progress of the theory of B-free subshifts in [16] and [19] and using [10], it is noticed in [21] that for each proximal B-free system there exists a (unique) taut B ′ -free subsystem such that F B ′ ⊂ F B and the density of the difference of these two sets vanishes, which "justifies" the conclusion that (X η , S ) is "relatively Behrend" over (X η ′ , S ) (in fact, the hereditary closures 5 of these two systems have the same sets of invariant measures). We should add that dynamics in the taut case is much better understood since it leads to the theory of hereditary 6 subshifts.…”

“…In both classes: proximal and minimal, the entropy can be positive and also zero. In fact, using [19], [16], it has been noticed in [21] that:…”