Tangencies in invariant disc packings for certain planar piecewise isometries are rare

Abstract

For planar piecewise isometries (PWIs) (two-dimensional maps that restrict to isometries on some partition) there is a natural coding given by the itinerary of a trajectory between the pieces (atoms) of the partition on which it is defined. The set of points with the same coding is referred to as a cell and under certain general conditions the periodically coded cells define an invariant set that is a disjoint union of discs. In this paper properties of this invariant disc packing are investigated. For a one-parameter family of PWI on a torus, it is proved that tangencies between discs in this packing are rare. More precisely it is shown, using algebraic constraints on the geometry of the centres of the discs, that tangencies between any two discs can only occur at a finite number of parameter values; hence all tangencies will occur at a set of parameter values that is (at most) countably infinite. If such packings are dense it can be shown that they are maximal in a sense of measure. Examples are provided to show that the packing may not be dense if there is continuity over boundaries in the partition, and also that the absence of tangencies in the packing does not necessarily imply that the complement of the packing has positive Lebesgue measure.