The Moment Hierarchy for Iterated Function Systems: Discovery and Detection of Invariant Measures from Data

Overview & Synopsis

Invariant measures characterize the long-term statistical behaviour of dynamical systems and are particularly central in analyzing stochastic and contractive dynamics. This paper introduces a data-driven convex optimization framework for discovering invariant measures of iterated function systems (IFS), including uniquely ergodic systems and those that are not. Leveraging the structure of the Hutchinson operator and polynomial test functions, we construct a semidefinite hierarchy to enforce moment-based invariance conditions from empirical data. For contractive, uniquely ergodic IFS, we prove that the empirical moment sequences converge to those of the unique invariant measure and that our convex relaxation hierarchy recovers increasingly accurate polynomial approximations. Crucially, when ergodicity fails-either due to lack of contraction or the presence of multiple invariant measures-we show that our method exhibits non-convergence, inconsistency across initializations, or structural degeneracy in the moment hierarchy. These behaviours serve as statistical certificates of non-uniqueness. Numerical examples, including affine and nonlinear IFS, illustrate the discovery of smooth invariant densities and the detection of bifurcation-induced non-ergodicity. Our results bridge classical IFS theory with modern data-driven operator-theoretic methods and provide a rigorous framework for understanding the distributional dynamics of random systems from data alone.