PERIODIC-ORBITS IN K-SYMMETRICAL DYNAMICAL-SYSTEMS

A map L is called k-symmetric if its kth iterate L(k) possesses more s ymmetry than L, for some value of k. In k-symmetric systems, there exi sts a notion of k-symmetric orbits. This paper deals with k-symmetric periodic orbits. We derive a relation between orbits that are k-symmet ric with respect to reversing k-symmetries and symmetric orbits of L(k ). With this relation we set out an efficient method for finding syste matically all periodic orbits that are k-symmetric with respect to rev ersing k-symmetries. This k-symmetric fixed set iteration (FSI) method generalizes a celebrated method due to DeVogelaere that applies to sy mmetric periodic orbits in reversible dynamical systems. We use the FS I method to study k-symmetric periodic orbits of a map of the plane R( 2) possessing a crystallographic reversing k-symmetry group. The expli cit findings illustrate a typically k-symmetric phenomenon, consisting of a nontrivial relation between the symmetry properties of periodic orbits and their periods.