Rudolph’s two step coding theorem and Alpern’s lemma for ℝ^{𝕕} actions
Bryna Kra, Anthony Quas, Ayşe Şahı̇n · Transactions of the American Mathematical Society · 2014
Rudolph showed that the orbits of any measurable, measure preserving R d \mathbb {R}^d action can be measurably tiled by 2 d 2^d rectangles and asked if this number of tiles is optimal for d > 1 d>1 . In this paper, using a tiling of R d \mathbb {R}^d by notched cubes , we show that d + 1 d+1 tiles suffice. Furthermore, using a detailed analysis of the set of invariant measures on tilings of R 2 \mathbb {R}^2 by two rectangles, we show that while for R 2 \mathbb {R}^2 actions with completely positive entropy this bound is optimal, there exist mixing R 2 \mathbb {R}^2 actions whose orbits can be tiled by 2 tiles.