Information Geometry of k-Exponential Families: Dually-Flat, Hessian and Legendre Structures

In this paper, we present a review of recent developments on the k-deformed statistical mechanics in the framework of the information geometry. Three different geometric structures are introduced in the k-formalism which are obtained starting from three, not equivalent, divergence functions, corresponding to the k-deformed version of Kullback-Leibler, "Kerridge" and Brègman divergences. The first statistical manifold derived from the k-Kullback-Leibler divergence form an invariant geometry with a positive curvature that vanishes in the k->0 limit. The other two statistical manifolds are related to each other by means of a scaling transform and are both dually-flat. They have a dualistic Hessian structure endowed by a deformed Fisher metric and an affine connection that are consistent with a statistical scalar product based on the k-escort expectation. These flat geometries admit dual potentials corresponding to the thermodynamic Massieu and entropy functions that induce a Legendre structure of k-thermodynamics in the picture of the information geometry.