The Emptiness Jump

In classical complex dynamics, the Julia set is fixed: it’s a property of the map, determined once and for all. Iterate z² + c and the Julia set is there — fractal, invariant, eternal.

Endo extends Julia-Fatou theory to random dynamical systems with complete connections, where the map applied at each step depends on both the current state and the system’s history. In this setting, each state has its own kernel Julia set, and these sets can do something impossible in the classical theory: they can spontaneously become empty along admissible trajectories.

Endo calls these “emptiness jumps.” A kernel Julia set is nonempty at one state — dynamics are chaotic there, with sensitive dependence on initial conditions. Along an admissible trajectory, the Julia set at the next state is suddenly empty — dynamics are equicontinuous, tame, predictable. Chaos evaporates in a single step.

The phenomenon has no classical analogue because classical Julia sets are properties of a single fixed map. In the random setting, the map changes with each step, and the interaction between state-dependent maps and history-dependent selection rules creates structural possibilities that fixed iteration cannot. The Cooperation Principle I shows that when kernel Julia sets are empty everywhere, the adjoint operator iterates exhibit equicontinuity on probability measures — global regularity emerges from local emptiness.

Conditions that exclude emptiness jumps include discrete state spaces and phi-irreducibility. The jumps live in continuous, non-Markovian territory — precisely where classical tools lose their grip.