Unique Ergodicity in Networks of Interacting Iterated Function Systems: Theory and Control

<div xmlns="http://www.tei-c.org/ns/1.0"> We present a novel framework for analyzing the ergodic properties of networks of interacting iterated function systems (IFS) with state-dependent probabilistic dynamics. Each node in the network evolves according to an IFS defined on a common Polish state space, with local randomness governed by position-dependent switching laws. The nodes are coupled through a graph, and their interactions influence the update rules, introducing a rich interplay between local randomness and global structure. We introduce the notion of joint unique ergodicity for such systems, defined as a unique invariant probability measure on the joint state space to which the empirical distribution of the entire network converges, independent of initial conditions. We characterize conditions under which coupling-through topology or dynamics-restores ergodicity, even when individual IFSs are nonergodic. Sufficient conditions are provided via Lyapunov drift, graph connectivity, and Wasserstein contraction over deterministic or random communication blocks. Our results extend to time-varying, asynchronous, and non-compact settings. </div>