Low complexity Alpern Lemmas: How can you tile a floor with a mixture of metric and imperial tiles?

Abstract:

In Ergodic Theory, one is often interested in constructing maps between pairs of systems. A common approach to this is based on ``cutting orbits into pieces". These pieces should ideally be simply described.

I will describe some theorems of this type, indicating how they are used and talk about recent work with Bryna Kra and Ayse Sahin, where we study the case of actions of $\mathbb{R}^d$.

Along the way, we discuss the possible tilings of an infinite plane with 1ft × 1ft tiles; and with 1m × 1m tiles.