Maps on statistical manifolds exactly reduced from the Perron-Frobenius equations for solvable chaotic maps

TL;DR

This paper derives exact maps on statistical manifolds from Perron-Frobenius equations for a family of chaotic maps, linking dynamical systems with information geometry and symplectic structures.

Contribution

It introduces a novel reduction of Perron-Frobenius equations to maps on parameter spaces, revealing geometric structures in chaotic maps.

Findings

Relations between statistical and orbital pictures established

Derived maps characterized by information geometric properties

Symplectic structures induced from statistical structures

Abstract

Maps on a parameter space for expressing distribution functions are exactly derived from the Perron-Frobenius equations for a generalized Boole transform family. Here the generalized Boole transform family is a one-parameter family of maps where it is defined on a subset of the real line and its probability distribution function is the Cauchy distribution with some parameters. With this reduction, some relations between the statistical picture and the orbital one are shown. From the viewpoint of information geometry, the parameter space can be identified with a statistical manifold, and then it is shown that the derived maps can be characterized. Also, with an induced symplectic structure from a statistical structure, symplectic and information geometric aspects of the derived maps are discussed.