Geodesics and dynamical information projections on the manifold of Hölder equilibrium probabilities
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We consider here the discrete time dynamics described by a transformation T:M → M, where T is either the action of shift T=σ on the symbolic space M={1,2,...,d}^ℕ, or, T describes the action of a d to 1 expanding transformation T:S^1 → S^1 of class C^1+α ( for example x → T(x) =d x (mod 1) ), where M=S^1 is the unit circle. It is known that the infinite-dimensional manifold 𝒩 of equilibrium probabilities for Hölder potentials A:M →ℝ is an analytical manifold and carries a natural Riemannian metric associated with the asymptotic variance. We show here that under the assumption of the existence of a Fourier-like Hilbert basis for the kernel of the Ruelle operator there exists geodesics paths. When T=σ and M={0,1}^ℕ such basis exists. In a different direction, we also consider the KL-divergence D_KL(μ_1,μ_2) for a pair of equilibrium probabilities. If D_KL(μ_1,μ_2)=0, then μ_1=μ_2. Although D_KL is not a metric in 𝒩, it describes the proximity between μ_1 and μ_2. A natural problem is: for a fixed probability μ_1∈𝒩 consider the probability μ_2 in a convex set of probabilities in 𝒩 which minimizes D_KL(μ_1,μ_2). This minimization problem is a dynamical version of the main issues considered in information projections. We consider this problem in 𝒩, a case where all probabilities are dynamically invariant, getting explicit equations for the solution sought. Triangle and Pythagorean inequalities will be investigated.
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