Duality and the Computation of Approximate Invariant Densities for Nonsingular Transformations

Abstract

We investigate a class of optimization problems which arise in the approximation of invariant densities for a nonsingular, measurable transformation T acting on a finite measure space. The problems under consideration have convex integral-type objectives and finite moment constraints and include, for example, the maximum entropy and quadratic programming approaches previously studied in the literature. This article is a natural sequel to those investigations and to the paper [C. Bose and R. Murray, Discrete Contin. Dyn. Syst., 14 (2006), pp. 597–615], where a general class of convergent moment approximations were defined such that the limiting optimal solution is an invariant density for T. This article mainly concerns the solution of a single finite moment problem arising from this general approximation scheme. Both theoretical aspects and computational issues are treated. Although the problem fits easily into the standard theory of duality in convex optimization, its dynamical origins lead to technical obstructions in the derivation of optimality conditions. In particular, the dual functional for our problem is neither strictly convex nor coercive, relating in part to the fact that the moment generating functions for the approximation scheme need not be pseudo-Haar. The method of the paper circumvents these obstructions and yields an unexpected benefit: each finite moment approximation leads to rigorous bounds on the support of all invariant densities for T.