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Rudolph showed that the orbits of any measurable, measure preserving $\mathbb R^d$ action can be measurably tiled by $2^d$ rectangles and asked if this number of tiles is optimal for $d>1$. In this paper, using a tiling of $\mathbb R^d$ by notched cubes, we show that $d+1$ tiles suffice. Furthermore, using a detailed analysis of the set of invariant measures on tilings of $\mathbb R^2$ by two rectangles, we show that while for $\mathbb R^2$ actions with completely positive entropy this bound is optimal there exist mixing $\mathbb R^2$ actions whose orbits can be tiled by 2 tiles.

Rudolph’s two step coding theorem and Alpern’s lemma for ℝ^{𝕕} actions