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

TL;DR

This paper improves the understanding of tiling orbits of measure-preserving  actions, reducing the number of tiles needed from 2^d to d+1, and explores the optimality of this bound in different entropy contexts.

Contribution

It introduces a new tiling method using notched cubes to show that d+1 tiles suffice for  actions, and analyzes invariant measures to determine bounds' optimality.

Findings

d+1 tiles suffice for  actions using notched cubes

Optimality of the tile bound depends on entropy conditions

Existence of mixing  actions with only 2 tiles

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.