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We highlight a few recent results on the effect of the diffusion process indeterministic area preserving maps with noncompact phase space, namely thestandard map. In more detail, we focus on the anomalous diffusion arising dueto the accelerator modes, i.e. resonant-like features of the phase space whichare transported in a super-diffusive (ballistic) manner. Their presence affectsalso the trajectories lying in the immediate neighborhood resulting inanomalous (non-Gaussian) diffusion. We aim to shed some light on these specialproperties of the phase space by utilizing the diffusion exponent (rate ofdiffusion) and the momentum distribution in terms of the L\'evy stabledistributions, with the goal to reach an understanding of the global behaviour.To this end we consider a rather large interval for the kick parameter ($0\le K\le 70$) of the standard map where accelerator modes of different periodicityexist.

Accelerator modes and their effect in the diffusion properties in the kicked rotator