  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).
