Discrete rotations and generalized continued fractions

The qualitative and experimental investigation of a dynamical system representing a discretization of a rotation is established. For different values of the angle of rotation and different initial data we found the first periodic point, the period, the maximal and minimal values of the coordinates of the trajectory points. The main result we found was the discovery of such a value of the angle of rotation for which all the trajectories with different initial positions (points) go very far from their initial positions and from the origin. The first periodic point occurs after more than 1500000 non-periodic points of the trajectory appear and the maximal value of its coordinates is greater than 1022. The value of the period is 2049. The angle for which such a phenomenon takes place differs from $\pi/2$ only at the 14th decimal place in the computer representation of $\pi/2$. The algebraic and algorithmic structure of the discrete rotation map is covered by the concept of the new theory of generalized continued fractions.