This paper presents a complete proof of a conj~ given by Ashwin, Dearie and Fu that the map describing the dynamical behavior of the Sigma-Delta modulator has a global attractor. By viewing the map as a piecewise rotation, and by geometric analysis, the authors give a simpler and more sufficient proof of the conjectm'e, than the one presented by ~e and published in Dynam/ca/ Systems, 2002,17: 377 -388. Key words piecewise rotation, global attractor, the first relaLm map. MSC 2e00 37E99

Iteration of a planar piecewise isometry may generate an invariant disk packing, and understanding the properties of the disk packing is helpful for estimating the Lebesgue measure of the exceptional set for the planar piecewise isometry. If the disk packing is not dense, then the Lebesgue measure of the exceptional set is positive. But it is not easy to check the density of a disk packing. In this paper, the authors present necessary and sufficient conditions for the density of a general disk packing, and discuss some properties of disk packings for planar piecewise isometries.

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In this paper, we discuss the invariant measures for planar piecewise isometries. It is shown that the Hausdorff measure restricted to an almost invariant set with respect to the Hausdorff measure is invariant.