Linear and fractal diffusion coefficients in a family of one-dimensional chaotic maps

We analyse deterministic diffusion in a simple, one-dimensional setting consisting of a family of four parameter dependent, chaotic maps defined over the real line. When iterated under these maps, a probability density function spreads out and one can define a diffusion coefficient. We look at how the diffusion coefficient varies across the family of maps and under parameter variation. Using a technique by which Taylor-Green-Kubo formulae are evaluated in terms of generalized Takagi functions, we derive exact, fully analytical expressions for the diffusion coefficients. Typically, for simple maps these quantities are fractal functions of control parameters. However, our family of four maps exhibits both fractal and linear behaviour. We explain these different structures by looking at the topology of the Markov partitions and the ergodic properties of the maps.