  Inspired by the algorithm of Barnsley's chaos game, we construct an open
quantum system model based on the repeated interaction process. We shown that
the quantum dynamics of the appropriate fermionic/bosonic system (in
interaction with an environment) provides a physical model of the chaos game.
When considering fermionic operators, we follow the system's evolution by
focusing on its reduced density matrix. The system is shown to be in a Gaussian
state (at all time $t$) and the average number of particles is shown to obey
the chaos game equation. Considering bosonic operators, with a system initially
prepared in coherent states, the evolution of the system can be tracked by
investigating the dynamics of the eigenvalues of the annihilation operator.
This quantity is governed by a chaos game-like equation from which different
scenarios emerge.
