Shaping of invariant measures of iterated function systems

Abstract

We study an inverse problem for iterated function systems: given maps $w_1,\dots,w_m$ on a Polish space and a target probability measure $ν$, choose the law $μ$ with which the maps are sampled so that the invariant measure $π_μ$ of the resulting Markov chain is as close as possible to $ν$ in the Wasserstein-1 distance. The minimization runs over the admissible set $A$ of sampling laws for which the chain has a unique invariant measure attracting every initial law; the maps need not be contractions. On bounded state spaces, regions certified by a Doeblin or splitting condition are convex unions of relatively open faces of the simplex, and they exhaust $A$ when the state space is finite; for affine systems with nonnegative matrices $A$ is a spectral sublevel set; in general $A$ is neither open, closed, convex nor connected. Existence of an optimal law reduces to the invariant measures left behind at the missing boundary $\mathrm{cl}(A)\setminus A$: boundary points without invariant measures of finite first moment repel minimizing sequences, which yields existence for all systems of homotheties of $\mathbb R^d$, and for compact state spaces a solution exists for every target if and only if the set of reachable invariant measures is closed; a two-state example shows that existence can fail. A search over a $δ$-net returns an $ε$-optimal law over any region on which $μ\mapstoπ_μ$ is Lipschitz and the chains are uniformly attracting, with explicit constants and, on an interval, an explicit sample size. Under contraction on average, $μ\mapstoπ_μ$ is differentiable, its derivative is injective if and only if the push-forwards $(w_i)_\#π_μ$ are linearly independent, and the objective is weakly convex, so that a projected subgradient method with inexact subgradients converges to near-stationary points.