Class renormalization: Islands around islands

Abstract

A ‘‘class’’-c orbit is one that rotates around a periodic orbit of class c-1 with some definite frequency. This contrasts with the ‘‘level’’ of a periodic orbit which is the number of elements in the continued-fraction expansion of its frequency. Level renormalization is conventionally used to study the structure of quasiperiodic orbits. The scaling structure of periodic orbits encircling other periodic orbits in area-preserving maps is discussed here. Renormalization fixed points of p/q bifurcations are found and scaling exponents determined. Fixed points for q>2 correspond to self-similar islands around islands. Frequencies of the island boundary circles at the fixed points are obtained. Importance of this scaling for the motion of particles in stochastic regions is emphasized.