On the discrete bicycle transformation

Abstract

We study the dynamics of the discrete bicycle (Darboux, Backlund) transformation of polygons in n-dimensional Euclidean space. This transformation is a discretization of the continuous bicycle transformation, recently studied by Foote, Levi, and Tabachnikov. We prove that the respective monodromy is a Moebius transformation. Working toward establishing complete integrability of the discrete bicycle transformation, we describe the monodromy integrals and prove the Bianchi permutability property. We show that the discrete bicycle transformation commutes with the recutting of polygons, a discrete dynamical system, previously studied by V. Adler. We show that a certain center associated with a polygon and discovered by Adler, is preserved under the discrete bicycle transformation. As a case study, we give a complete description of the dynamics of the discrete bicycle transformation on plane quadrilaterals.