A piecewise quadratic maximum entropy method for the statistical study of chaos

Let the Frobenius–Perron operator PS:L1(0,1)→L1(0,1)PS:L1(0,1)→L1(0,1), related to a nonsingular transformation S:[0,1]→[0,1]S:[0,1]→[0,1], have an invariant density f⁎f⁎. We propose a piecewise quadratic maximum entropy method for the numerical approximation of f⁎f⁎. The role of the partition of unity property of the quadratic functions for the numerical recovery of f⁎f⁎ has been depicted. The proposed algorithm overcomes the ill-conditioning shortage of the traditional maximum entropy method which only employs polynomials, so that any number of moments can be used to increase the accuracy of the computed invariant density. The convergence rate of the method is shown to be of order 3. Numerical results are presented to justify the theoretical analysis of the method.