  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.
