Dynamical Ordering of Non-Birkhoff Orbits and Topological Entropy in the Standard Mapping

Abstract

The standard mapping is an analytical, reversible monotone twist mapping. The appearance ordering (i.e. the so-called dynamical ordering), of symmetric non-Birkhoff periodic orbits (SNBO) in the standard mapping is derived. Essential use is made of the reversibility. After the establishment of various properties of the symmetry axes under the mapping, two theorems connecting the dynamical ordering are proved. Then, the braids for SNBOs are constructed with the aid of techniques developed in braid group theory. A lower bound of the topological entropy of a system possessing an SNBO is obtained using the eigenvalue of the reduced Burau matrix representation of the braid constructed from the SNBO. The behavior of the topological entropy in the integrable limit is discussed.