Uniqueness and ergodic properties of attractive g-measures

Abstract

We consider g-measures for the shift on 11^=0 $ where S is a finite set. For a certain class of continuous g, two g-measures are identified; they are equal if and only if there is a unique g-measure. We prove that the natural extensions of these measures are Bernoulli. With a further restriction on g when S is a two-point set, we show that there is a unique g-measure. We also consider extensions of these results to the non-continuous case.

Explicit estimates in the Bramson–Kalikow model

“…The proof follows from the properties of attractive functions g ∈ G and associated g-measures described in Hulse (2006Hulse ( , 1991.…”

“…As in Hulse (1991), for x, y ∈ X , a, b ∈ A, and h 1 , h 2 ∈ C(X ), we define the function P : X × X → [0, 1] by P (ax, by) = min {h 1 (ax), h 2 (ay)} if a = b {h 1 (ax) − h 2 (ay)} ∨ 0 otherwise, and a∈A b∈A P (ax, by) = 1. Let f ∈ C(X ).…”

“…For any functions f 1 , f 2 ∈ C(X ), we have that lim n→∞ L n P (f 1 ⊗ f 2 )(+1, +1) exists and defines a coupling ν between ν + 1 and ν + 2 (Hulse (1991)). By construction and definition of P , this coupling has the property that ν({(x, y) ∈ X × X : x 0 < y 0 }) = 0.…”

“…Notwithstanding, the problem of non-uniqueness is much less understood and the literature is still based on few examples (Bramson & Kalikow, 1993;Hulse, 2006;Berger et al, 2005). As far as we know, general criteria for non-uniqueness have only been obtained for the class of regular attractive functions (Gallo & Takahashi, 2013;Hulse, 1991).…”

“…Instead, we study the properties of a sequence of suitably chosen Markov chains. Theorem 2 is inspired by the works of Bramson & Kalikow (1993), Lacroix (2000), and Hulse (1991), but has the advantage of being formulated using thed-distance, which is key to our constructive proof of Theorem 1. Moreover, Theorem 3 states that our criterion (Theorem 2) is optimal in the important class of binary regular attractive functions, giving a necessary and sufficient condition for non-uniqueness in this class, which includes the BK model.…”