Julia sets are the boundaries of chaos in complex dynamics. For a polynomial or rational map, the Julia set is where iterates are sensitive to initial conditions — the fractal frontier between orbits that escape and orbits that remain bounded. The Julia set can be a complicated curve, a Cantor dust, or the entire Riemann sphere, but it is always there. Non-empty Julia sets are a structural feature of deterministic iteration.

In random systems with complete connections, the Julia set can vanish.

The mechanism is "emptiness jumps" — a phenomenon unique to non-Markovian random dynamics. In a random system with complete connections, the rule selecting which map to apply at each step depends on the entire history, not just the current state. This history dependence introduces feedback: the system's trajectory influences its own future dynamics. Along admissible state trajectories, the kernel Julia set — the Julia set associated with a particular realization of the random process — can drop from non-empty to empty discontinuously.

The Cooperation Principle captures the global consequence: if the kernel Julia set is empty at every state and the admissible maps are open, then the iterates of the adjoint transition operator are equicontinuous on the entire space of probability measures. The chaos boundary doesn't gradually thin — it disappears, and when it does, the system becomes uniformly well-behaved.

This is a genuine novelty of history-dependent dynamics. In Markovian random dynamical systems, the Julia set may shrink or change shape, but it doesn't jump to empty along trajectories. The history dependence creates a mechanism for the system to cooperatively suppress its own chaotic boundary — the accumulated trajectory information constructively eliminates the sensitivity that individual maps would preserve.

The chaos was always conditional. History-dependent dynamics can revoke the condition.