A Decomposition of the Rotation of Circle

Abstract

The paper concerns the dynamics-related properties of the rotation map of a circle (rotation of the plane). A self-similar structure of orbits of the rotation map is established. That is, a possibility of decomposition of orbits of a given rotation map into a finite set of orbits of other such maps is proved—it is shown that every orbit of iterates of the rotation of circle on irrational angle, after linear re-scaling of its argument can be represented as a finite set of such orbits situated on another circles. A pointwise self-similarity of classical trigonometric system is established and an application to Fourier expansion, which emphasizes a possibility of shifting of signals with respect to time, is presented. The free mechanical motion is also considered. A special dynamical spectrum of frequencies or speeds, associated with a given uniform circular or rectilinear motion, is defined. We prove that an appropriate fragmentation of time axis yields a decomposition of a given orbit of the free continuous-time motion into a set of such orbits propagating in new time and such decomposition is consistent with the decomposition of the per time unit discrete motion. Particularly, our theorems assert that due to a piecewise-linear transform of spatial and time variables the rectilinear rays change their direction.