Reversing Symmetries in a Two-Parameter Family of Area-Preserving Maps

Abstract

The integrable twist map is used to derive an invertible two-parameter family of areapreserving maps specified by two arbitrary functions of a real variable. It has been shown that the integrable case has infinitely many reversing symmetries (not necessarily all involutory). Almost all of these symmetries are destroyed under addition of non-integrable terms. However some involutory reversing symmetries survive if certain restrictions on the functions are imposed. An involutory reversing symmetry exists in three different cases. In the first two cases, the symmetry lines are continuous curves leading to infinitely many symmetric periodic orbits. The third case is an interesting example of an involutory reversing symmetry in which the corresponding symmetry lines are isolated points leading to none or few symmetric periodic orbits. The combined restrictions of the first two cases lead to a family of maps with double reversing symmetries. Furthermore there exists a subset of this family (identified by an additional restriction) with quadruple reversing symmetries.