Self-similar example of a piecewise isometric attractor

We describe an attracting piecewise rotation with two atoms with a self similar structure of periodic domains on its attractor which resembles Sierpiriski's gasket. Besides its natural beauty, this example appears a return map in certain piecewise affine maps on the torus which have been studied by Adler, Kitchens, and Tresser, and by a number of other researchers advancing the theory of digital filters.