Strong Anomaly in Diffusion Generated by Iterated Maps

We investigate the diffusion generated deterministically by periodic iterated maps that are defined by xₜ₊₁0ex0ex=0ex0exxₜ+axₜᶻexp[-(b/xₜ)^z-1], $z>1$. It is shown that the obtained mean squared displacement grows asymptotically as σ²(t)~ln^1/(z-1)(t) and that the corresponding propagator decays exponentially with the scaling variable |x|/√σ²(t). This strong diffusional anomaly stems from the anomalously broad distribution of waiting times in the corresponding random walk process and leads to a behavior obtained for diffusion in the presence of random local fields. A scaling approach is introduced which connects the explicit form of the maps to the mean squared displacement.