Deterministic diffusion in dynamical systems with a tiled phase space

TL;DR

The paper proves normal deterministic diffusion for dynamical systems with tiled phase spaces in both 1D and 2D. It employs the Perron–Frobenius operator and Fourier analysis of convolution kernels to show that, under suitable tilings and linear lifting maps, evolving densities converge to Gaussian forms with explicit diffusion coefficients: D = (Λ^2−1)/24 in 1D tilings and D = 5/4 in the 2D triangular tiling with Λ = 4. It further extends the results to odd Λ in hexagon tilings, establishing that deterministic diffusion is robust under these tiled geometries. These findings connect deterministic chaotic dynamics with classical diffusion behavior and provide concrete rates of spread for tiled-phase-space systems.

Abstract

The existence of normal deterministic diffusion in dynamical systems with a two-dimensional phase space tiled by regular triangles (or their unions into regular hexagons) is proven.