Fractal diffusion coefficients in simple dynamical systems

Deterministic diffusion is studied in simple, parameter-dependent dynamical systems. The diffusion coefficient is often a fractal function of the control parameter, exhibiting regions of scaling and self-similarity. We will give an overview of some analytical results and compare different approximating techniques. In particular, parameter-dependent diffusion coefficients are analytically derived in simple cases by solving the Taylor-Green-Kubo formula via a recursion relation solution of fractal ‘generalised Takagi functions’. The practicability of different techniques for approximating the diffusion coefficient, as well as their capability in exposing a fractal structure is compared. Analytical investigation into the dependence of the diffusion coefficient on the size and position of escape holes is undertaken. An exploration of a method which involves evaluating the zeros of a system’s dynamical zeta function via the weighted Milnor-Thurston kneading determinant is performed.