Dynamical behaviors and codings of invertible planar piecewise isometric systems

In this paper, the authors discuss some important sets, including singular sets, exceptional sets, irrational sets, etc., generated by invertible planar piecewise isometric maps. They first extend the coding map onto the entire phase space and characterize singular sets and exceptional sets by the number of codings. Then they reveal the relation among these sets, and show that an irrational set is contained in an exceptional set. And they confirm the existence of admissible rational codings under some conditions (including a necessary and sufficient condition and a sufficient condition). They also discuss dynamical properties of the discontinuity line segments which determine the dynamics of a piecewise isometric system. Finally, they list some open problems for further research. (C) 2009 Elsevier Ltd. All rights reserved.