Rudolph's Two-Step Coding Theorem and Alpern's Lemma for R^d Actions

Abstract

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.

arXiv commentVersion to appear in TAMS. Typo in title corrected

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