  We study periodic, piecewise linear maps on the plane starting with the Mort
Brown's map. We show that if the number of pieces is two, there is only a short
list of possible periods (this fact can be seen as the crystallographic
restriction for this class of maps). Otherwise, without the restriction on the
number of pieces, a map can have any period. We show how to construct such maps
using binary trees and so called admissible sequences.
