2007/12/17 by François Huveneers, Huveneers, François · 1 citation
Mathematics · Physics and Astronomy · #Mathematical Dynamics and Fractals #Advanced Mathematical Theories and Applications #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.0712.2731
The origin of deterministic diffusion is a matter of discussion. We study the asymptotic distributions of the sums yn(x)=∑k=0n-1ψ(x+kα), where ψ is a periodic function of bounded variation and α an irrational number. It is known that no diffusion process will be observed. Nevertheless, we find a picewise constant function ψ and an increasing sequence of integer (nj)j such that the limit distribution of the sequence (ynj/√ j)j is Gaussian (with stricly positive variance). If α is of constant type, we show that the sequence (nj)j may be taken to grow exponentially (this is close to optimal in some sense, and one has ||ynj||\mathrm L2∼ max0≤ k≤ nj||yk||\mathrm L2 as j→∞). We give an heuristic link with the theory of expanding maps of the interval.