A random dynamical system applies a sequence of maps chosen at random. In the simplest case, the choice is i.i.d. — each map is selected independently. But in a system with complete connections, the choice of the next map depends on the current state. The system has memory: where you are influences which map acts next.
The paper on polynomial random dynamical systems with complete connections (arXiv: 2603.20653) studies the probability that a random orbit on the Riemann sphere escapes to infinity.
On the Fatou set — the region of stable dynamics — the escape probability is locally constant: nearby points have the same probability of escaping. When all kernel Julia sets are empty (a non-degeneracy condition), the escape probability is continuous everywhere. The state-dependent reinforcement can create discontinuities that vanish under truncation, and stationary averaging produces genuine mixed behavior — escape probabilities that are positive but less than one everywhere.
The through-claim: memory in the dynamics creates escape probabilities that are neither 0 nor 1. In memoryless (i.i.d.) random dynamics, the escape probability is typically determined by the Fatou–Julia decomposition — either you escape or you don't. State-dependent selection blurs this dichotomy: the reinforcement can maintain orbits in an intermediate state where escape is uncertain, and the uncertainty is intrinsic, not due to initial conditions.
2603.20653. Dynamical systems / random iteration / polynomial dynamics / Riemann sphere / escape probability.