The role of ergodicity and mixing in the central limit theorem for Casati–Prosen triangle map variables

In this Letter we analyse the behaviour of the probability density function of the sum of N deterministic variables generated from the triangle map of Casati–Prosen. For the case in which the map is both ergodic and mixing the resulting probability density function quickly concurs with the Normal distribution. However, when the map is weakly chaotic, and fuzzily not mixing, the resulting probability density functions are described by power-laws. Moreover, contrarily to what it would be expected, as the number of added variables N increases the distance to Gaussian distribution increases. This behaviour goes against standard central limit theorem. By extrapolation of our finite size results we preview that in the limit of N   going to infinity the distribution has the same asymptotic decay as a Lorentzian (or a q=2q=2-Gaussian).