Arcwise connectedness of the set of ergodic measures of hereditary shifts

Abstract

We show that the set of ergodic invariant measures of a shift space with a safe symbol (this includes all hereditary shifts) is arcwise connected when endowed with the d-bar metric. As a consequence the set of ergodic measures of such a shift is also arcwise connected in the weak-star topology and the entropy function over this set attains all values in the interval between zero and the topological entropy of the shift (inclusive). The latter result is motivated by a conjecture of A. Katok.A shift space X over the alphabet Λ = {0, 1, . . . , n − 1} for some n ≥ 2 is hereditary if x ∈ X and y ≤ x (coordinate-wise) imply y ∈ X. Hereditary shifts were introduced by Kerr and Li in [20, p. 882] and their basic properties are presented in [25]. We say that a ∈ Λ is a safe symbol for a shift space X ⊂ Λ Z (see [31]) if for every z ∈ Λ Z obtained by replacing some entries in y ∈ X by a, we have that z ∈ X. By definition 0 is a safe symbol for every hereditary shift.The family of hereditary shifts includes: spacing shifts, beta shifts, bounded density shifts, B-admissible shifts; also, many examples of B-free shifts and some shifts of finite type are hereditary. All classes on that list have been extensively studied, with the B-free shifts attracting much attention recently (see Section 4 for more details). In our setting of Z actions shifts with a safe symbol seem to be less important. The notion is useful in the context of higher dimensional shifts (Z d actions with d ≥ 2, see [31] and references therein). It is easy to find examples of shift spaces over {0, 1, 2} which have 0 as a safe symbol, but are not hereditary.It should be no surprise that there are very few theorems applicable to all members of such a diverse family of shift spaces. Nevertheless, the main result of this note implies that there is a common feature of all hereditary shift spaces: for any hereditary shift X and for every t ≥ 0 the set of ergodic invariant measures with entropy less than or equal t, denoted M (t) σ (X) and endowed with the d-bar metric d M is arcwise connected (Theorem 6). Our proof shows that it is enough to assume that there exists a safe symbol, and actually shows that there is ad Ω -continuous arc in X consisting of generic points for ergodic measures from the arc in M (t) σ (X). This is a single orbit result very much in the spirit of [40]. Hered Ω is a pseudometric on X given by the upper asymptotic density of the set of indices two sequences in X differ.The d-bar metricd M induces a stronger topology than the usual weak * topology on the space of ergodic invariant measures. It follows that in the latter topological space the set of ergodic invariant measures with entropy less than or equal t is

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