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.
Heteroclinic bifurcations in four dimensional vector fields are investigated by setting up a local coordinates near a rough heteroclinic loop. This heteroclinic loop has a principal heteroclinic orbit and a non-principal heteroclinic orbit that takes orbit flip. The existence, nonexistence, coexistence and uniqueness of the 1-heteroclinic loop, 1-homoclinic orbit and 1-periodic orbit are studied. The existence of the two-fold or three-fold 1-periodic orbit is also obtained.
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