Multifractal Properties of Discrete Stochastic Mappings

Abstract

In different fields of statistical physics, such as Id random field Ising models (RFIM) and neural networks, there appear discrete stochastic mappings of the form

$${x_n} = {f_n}({x_{n - 1}})$$

((1))

where f _n(x) is chosen with equal probability from two functions fσ(x), σ = + or -, and 0 < f′ _σ(x) ≤ 1. The dynamics is nonchaotic and, in a region of physical parameters, converges to a strange (fractal) attractor as may be visualized in Fig. 1. The mapping generates an invariant measure which undergoes, as parameters are changed, qualitative transitions, for instance, a transition from thin to fat multifractal.