Zeta measures and Thermodynamic Formalism for temperature zero

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We address the analysis of the following problem: given a real Hölder potential f defined on the Bernoulli space and μf its equilibrium state, it is known that this shift-invariant probability can be weakly approximated by probabilities in periodic orbits associated to certain zeta functions. Given a Hölder function f>0 and a value s such that 0<s<1, we can associate a shift-invariant probability νₛ such that for each continuous function k we have ∫ k dνₛ=∑ₙ₌₁∞∑x∈ Fixₙesfⁿ(x)-nP(f)kⁿ(x)n∑ₙ₌₁∞∑x∈ Fixₙesfⁿ(x)-nP(f), where P(f) is the pressure of f, Fixₙ is the set of solutions of σⁿ(x)=x, for any n∈ N, and fⁿ(x) = f(x) + f(σ(x)) + f(σ²(x))+... + f(σⁿ⁻¹ (x)). We call νₛ a zeta probability for f and s. It is known that νₛ → μf, when s → 1. We consider for each value c the potential c f and the corresponding equilibrium state μc f. What happens with νₛ when c goes to infinity and s goes to one? This question is related to the problem of how to approximate the maximizing probability for f by probabilities on periodic orbits. We study this question and also present here the deviation function I and Large Deviation Principle for this limit c→ ∞, s→ 1. We will make an assumption: → ∞, ₛ→ ₁ c(1-s)= L>0. We do not assume here the maximizing probability for f is unique.