Pathological Anomalous Diffusion Generated by a Generalized Shift Map

Abstract

The dynamics of a decimal point of two-sided infinite sequences (… a_t^(-2) a_t^(-1). a_t⁽⁰⁾ a_t⁽¹⁾ …) is studied, where the motion of the point is governed by a generalized shift (GS) map. Diffusive motion and fluctuated drift motion are observed which depend on the rule of GS. We introduce a parameter P which is the probability of a⁽ⁱ⁺¹⁾₀ whose value is the same with a⁽ⁱ⁾₀. We calculate the mean square displacement σ²(n) ∝ n^ζ for a fluctuated drift rule, and find that super-diffusive (ζ∼eq 1.96) and sub-diffusive (ζ∼eq 0.93) for P=0.5. Pathological anomalous diffusion (ζ > 2) is observed if P approaches to 1.