Numerical approximation of conditionally invariant measures via Maximum Entropy
Abstract
It is well known that open dynamical systems can admit an uncountable number of (absolutely continuous) conditionally invariant measures (ACCIMs) for each prescribed escape rate. We propose and illustrate a convex optimisation based selection scheme (essentially maximum entropy) for gaining numerical access to some of these measures. The work is similar to the Maximum Entropy (MAXENT) approach for calculating absolutely continuous invariant measures of nonsingular dynamical systems, but contains some interesting new twists, including: (i) the natural escape rate is not known in advance, which can destroy convex structure in the problem; (ii) exploitation of convex duality to solve each approximation step induces important (but dynamically relevant and not at first apparent) localisation of support; (iii) significant potential for application to the approximation of other dynamically interesting objects (for example, invariant manifolds).
 Publication:

arXiv eprints
 Pub Date:
 October 2012
 arXiv:
 arXiv:1211.0068
 Bibcode:
 2012arXiv1211.0068B
 Keywords:

 Mathematics  Dynamical Systems;
 37M25 65K10 49M29 94A17
 EPrint:
 8 Figures