We consider the dynamical system described by the area-preserving standard mapping. It is known for this system that $P\left(t\right)$, the normalized number of recurrences staying in some given domain of the phase space at time $t$ (so-called “survival probability”) has the power-law asymptotics, $P\left(t\right)\sim t^{-\nu }$. We present new semiphenomenological arguments which enable us to map the dynamical system near the chaos border onto the effective “ultrametric diffusion” on the boundary of a treelike space with hierarchically organized transition rates. In the framework of our approach we have estimated the exponent $\nu$ as $\nu =\text{ln}2/\text{ln}\left(1+r_g\right)\approx 1.44$, where $r_g=\left(\sqrt{5}-1\right)/2$ is the critical rotation number.

• Received 13 February 2010

DOI:https://doi.org/10.1103/PhysRevE.81.046211