Piecewise expanding maps on the plane with singular ergodic properties

For 1 less than or equal to r < <infinity>, we construct a piecewise C-r expanding map F : D --> D on the domain D = (0, 1) x (-1, 1) subset of R-2 with the following property: there exists an open set B in D such that the diameter of F-n(B) converges to 0 as n --> infinity and the empirical measure n(-1) Sigma (n-1)(k=0) delta (Fk(x)) converges to the point measure delta (p) at p = (0, 0) as n --> infinity for any point x is an element of B.