Birkhoff's ergodic theorem and the piecewise-constant maximum entropy method for Frobenius-Perron operators

摘要

Let (Χ, ∑, σ) be a finite measure space and 5 : Χ→Χ be a nonsingular transformation such that the corresponding Frobenius-Perron operator Ps : L~1(Χ) → L~1(Χ) has a stationary density f*. We propose a piecewise-constant maximum entropy method for the numerical recovery of f* and give its relation to the classic Birkhoff's individual ergodic theorem. An advantage of the piecewise-constant method over the current maximum entropy method based on polynomial basis functions is that a nonlinear system of equations is not needed for solving the related moment problem. Numerical results are given for several one dimensional test mappings.