Scaling Treatment of Critical and "Chaotic" Dynamics of the Dilute Heisenberg Chain

A lattice rescaling method is applied to the equations of motion of the dilute Heisenberg chain and leads via a probabilistic integral equation to an iterative map for the characteristic frequency $\beta$ and concentration $p$. Dilution induces a crossover in the $\beta$ scaling from "chaotic" (ergodic and mixing) behavior, corresponding to the sampling of the pure band, to periodic orbits corresponding to isolated cluster response. A dynamic scaling form is obtained for the critical dynamics by fixed-point analysis.