Dimensional reduction and universal f(α) in magnetic chains

We study both multivalued discrete and continuum spin chains in a binary random field. The discrete spin model with N values is mapped to a (N-1)-dimensional dynamical system. The system shows a drastic dimensional reduction with the important consequence that the local magnetization is a fractal. For small field, for the Ising and the continuum spin-O(n) chains, the multifractal properties of the magnetization can be described by a universal f(α) curve