On the measure of the structure around the last KAM torus before and after its break-up

Using mappings as model problem we study the structure around the last invariant KAM torus for different values of the perturbing parameter. We used the standard map far the analysis of the hierarchical structure existing around invariant KAM tori applying two complementary methods: the Laskar's frequency map analysis and the sup-map analysis. We recover and extend the theoretical prediction recently given by Morbidelli and Giorgilli about the existence of a neighborhood almost completely full of slave tori around a chief torus. We then make tests about the diffusion in order to measure the barrier to diffusion which still remains after the break-up of the last KAM torus.