Numerical approximation of conditionally invariant measures via Maximum Entropy

TL;DR

This paper introduces a convex optimization method based on maximum entropy to numerically approximate conditionally invariant measures in open dynamical systems, addressing challenges like unknown escape rates and support localization.

Contribution

It extends the maximum entropy approach to open systems, incorporating unknown escape rates and support localization, with potential applications to invariant manifolds.

Findings

Successfully approximates conditionally invariant measures using convex optimization.

Demonstrates the method's ability to handle unknown escape rates.

Shows potential for broader applications in dynamical systems analysis.

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).