Rudolph showed that the orbits of any measurable, measure preserving Rd action can be measurably tiled by 2drectangles and asked if this number of tiles is optimal for d > 1. In this paper, using a tiling of ℝdby notched cubes, we show that d + 1 tiles suffice. Furthermore, using a detailed analysis of the set of invariant measures on tilings of ℝ2by two rectangles, we show that while for ℝ2actions with completely positive entropy this bound is optimal, there exist mixing ℝ2actions whose orbits can be tiled by 2 tiles.
ASJC Scopus subject areas
- Applied Mathematics