Dynamics of symplectic piecewise affine elliptic rotation maps on tori

Abstract

We investigate the iterative dynamics of symplectic piecewise-affine elliptic rotation maps associated with matrices with non-integer entries, \big(\begin{smallmatrix} 0 & -1 \\ 1 & a \end{smallmatrix}\big), where -2 < a < 2. We explain how their singularity and periodicity structures develop in general. In a special case, a = \pm \sqrt 2, we completely determine the orbit structure, the singularity structure, the invariant fractal and the ergodicity on the invariant fractal with respect to its Hausdorff measure. We also report an example of a period tripling cascade that arises from this system.