The Ruelle operator for symmetric $\beta$-shifts

Abstract

Consider m ∈ N and β ∈ (1, m + 1]. Assume that a ∈ R can be represented in base β using a development in series a = ∞ n=1 x(n)β −n , where the sequence x = (x(n)) n∈N takes values in the alphabet Am := {0, . . . , m}. The above expression is called the β-expansion of a and it is not necessarily unique. We are interested in sequences x = (x(n)) n∈N ∈ A N m which are associated to all possible values a which have a unique expansion. We denote the set of such x (with some more technical restrictions) by X m,β ⊂ A N m . The space X m,β is called the symmetric β-shift associated to the pair (m, β). It is invariant by the shift map but in general it is not a subshift of finite type. Given a Hölder continuous potential A : X m,β → R, we consider the Ruelle operator L A and we show the existence of a positive eigenfunction ψ A and an eigenmeasure ρ A for some values of m and β. We also consider a variational principle of pressure. Moreover, we prove that the family of entropies (h(µ tA )) t>0 converges, when t → ∞, to the maximal value among the set of all possible values of entropy of all A-maximizing probabilities.

Existence of Gibbs states and maximizing measures on a general one-dimensional lattice system with markovian structure

“…which will be a necessary result in the proof of Proposition 2 below. By compactness of the set B(A, I), it is guaranteed existence of ϕ-maximizing measures associated to a potential ϕ ∈ H α (B(A, I)) as accumulation points in the weak* topology of the family of equilibrium states (µ tϕ ) t>1 , which are known as ground-states (see for instance [BLL13], [CGU11] and [LV20]). The above, is the main tool that we will use in the proof of Proposition 2.…”

“…)) , which implies, using (18), that µ ′ ∞ = lim n∈N µ φ ′ tn is a ϕ ′ -maximizing measure (see for instance [BLL13] and [LV20]). Thus, for any µ…”