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Scaling properties of discontinuous maps

We study the scaling properties of discontinuous maps by analyzing the average value of the squared action variable $I^2$. We focus our study on two dynamical regimes separated by the critical value $K_c$ of the control parameter $K$: the slow diffusion ($K<K_c$) and the quasilinear diffusion ($K>K_c$) regimes. We found that the scaling of $I^2$ for discontinuous maps when $K\ll K_c$ and $K\gg K_c$ obeys the same scaling laws, in the appropriate limits, than Chirikov&#39;s standard map in the regimes of weak and strong nonlinearity, respectively. However, due to absence of KAM tori, we observed in both regimes that $I^2\propto nK^β$ for $n\gg 1$ (being $n$ the $n$-th iteration of the map) with $β\approx 5/2$ when $K\ll K_c$ and $β\approx 2$ for $K\gg K_c$.