Singularities of two-dimensional invertible piecewise isometric dynamics.

We investigate the singularity structure of two-dimensional invertible piecewise isometric dynamics. The main purpose of this research is to establish the connection between the geometrical properties of the singularity and the dynamics of the system. We classify the singularity of two-dimensional bounded invertible piecewise isometric dynamics into three types with respect to their geometrical properties. Among the three, we show that one type of the singularity can be removed by lifting up the dynamics to a suitably defined (branched) manifold. Among the remaining two, we prove that only one of them contributes to the intricate orbit structure of the system and generates the sensitive dependence on the initial condition, while the other does nothing.