Finitary isomorphisms in high dimensions
A fundamental problem in ergodic theory is to classify those processes which are isomorphic to a Bernoulli shift (an IID process). While this problem is well understood in various settings (in particular, for processes on $\mathbb{Z}^d$ in any dimension $d$), the isomorphisms tend to be obtained via non-constructive methods. Finitary isomorphisms are a more constrained class of isomorphisms which tend to be constructed via explicit algorithms. The analogous finitary isomorphism classification problem is much less understood, with only partial results in one dimension, and no results at all in higher dimensions. I will discuss some ideas from a work in progress on finitary isomorphisms in $\mathbb{Z}^d$, focussing primarily on the probabilistic tools needed, which are of independent interest.