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

Abstract

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.

Keywords:

• Frobenius–Perron operator
• stationary density
• maximum entropy
• piecewise-constant approximations

2000 AMS Subject Classifications:

• 41A35
• 65D07
• 65J10

Acknowledgements

The authors are thankful for the helpful suggestions of anonymous referees which improved the paper, and for introducing the paper 11.