Hereditary subshifts whose measure of maximal entropy does not have the Gibbs property

Abstract

Consider a partial order on {0, 1} Z : x ≤ y when x i ≤ y i for all i ∈ Z. A subshift X ⊂ {0, 1} Z is hereditary if together with any x ∈ {0, 1} Z it contains all y ≤ x. Heuristically speaking, a hereditary subshift contains all the elements between maximal elements (with respect to this partial order) and the element 0 Z . In a particular situation when it suffices to take (the orbit closure of) all the elements between a single maximal element x and the element 0 Z , we speak of subordinate subshifts. In this paper we investigate measure-theoretic properties of such subshifts, with a special emphasis on thermodynamical formalism. The key notion is a measure-theoretic counterpart of subordinate subshifts, 4 Sandwich measure-theoretic subordinate subshifts 20 4.

On the Garden of Eden theorem for B-free subshifts

“…Now, notice that if ν η has full support, then also ν η ˚Bp 1 2 , 1 2 q has full support in r X η (cf. [30]) and therefore, in view of Corollary 3.9 and Theorems 2.2 and 2.3, we obtain the following.…”

“…Let us recall the following result, which is implicit in the proof of Eqn. (25) on page 13 in [30]. Its proof is a rather direct consequence of Theorem 2.3.…”

“…Moreover, they are hereditary: if x belongs to the subshift and if y ď x coordinatewise, then y belongs to the subshift [1]. On the other hand, the unique measure of maximal entropy is not Gibbs [30], a rather surprising property that was first discovered for the square-free subshift [38]. Moreover, their automorphism group is trivial, see below.…”

“…From a dynamical point of view, the role of tautness is seen in the following three results. The first one combines [30,Cor. 1.11] [30]).…”

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

“…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.…”