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

Let (X, Σ, σ) be a finite measure space and S: X→X be a nonsingular transformation such that the corresponding Frobenius–Perron operator P S : L 1(X)→L 1(X) 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.