Rate of convergence in the disjunctive chaos game algorithm.

Abstract

The rate of convergence of the chaos game algorithm for recovering attractors of contractive iterated function systems (IFSs) is studied. As with successive Picard iterates in the Banach fixed point principle, one has the exponential convergence. However, a symbolic sequence driving the iteration needs to obey some suitable statistical properties. Specifically, this sequence needs to behave like the classical Champernowne sequence. The exponent of convergence can be estimated from below in terms of (lower and upper) box dimensions of the attractor and from above by the entropy of the driver discounted by the Lipschitz constant of the IFS. Generically (in the sense of the Baire category), a driver that recovers the attractor yields arbitrarily slow convergence (of infinite order) interlaced with arbitrarily fast possible convergence (of order approaching a lower dimension).