An Elementary Approach to Rigorous Approximation of Invariant Measures

作者：Galatolo Stefano^*; Nisoli Isaia

来源：SIAM Journal on Applied Dynamical Systems, 2014, 13(2): 958-985.

DOI：10.1137/130911044

• approximation of invariant measure
• transfer operator
• fixed point approximation
• Lyapunov exponent
• interval arithmetics

We describe a framework in which it is possible to develop and implement algorithms for the approximation of invariant measures of dynamical systems with a given bound on the error of the approximation. Our approach is based on a general statement on the approximation of fixed points for operators between normed vector spaces, allowing an explicit estimation of the error. We show the flexibility of our approach by applying it to piecewise expanding maps and to maps with indifferent fixed points. We show how the required estimations can be implemented to compute invariant densities up to a given error in the L-1 or L-infinity distance. We also show how to use this to compute an estimation with certified error for the entropy of those systems. We show how several related computational and numerical issues can be solved to obtain working implementations and experimental results on some one dimensional maps.

• 出版日期2014