Chaotic Hamiltonian systems revisited: Survival probability

Abstract

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