Uniformly Expanding Random Walks on Manifolds

TL;DR

This paper develops a method to construct uniformly expanding random walks on smooth manifolds, extending previous results from tori to all closed manifolds of any dimension, with a focus on higher-dimensional subspace growth.

Contribution

It generalizes the construction of uniformly expanding random walks from tori to all closed manifolds of any dimension, broadening the scope of previous work.

Findings

Constructed uniformly expanding random walks on all closed manifolds.

Extended the concept of expansion to higher-dimensional subspaces.

Built a robust class of examples demonstrating these properties.

Abstract

In this paper we construct uniformly expanding random walks on smooth manifolds. In higher dimensions, our definition of uniform expansion measures the growth of subspaces rather than single vectors. Potrie showed that given any open set $U$ of $\text{Diff}^\infty_{vol}(\mathbb{T}^2)$, there exists an uniformly expanding random walk $\mu$ supported on a finite subset of $U$. In this paper we extend those results to closed manifolds of any dimension, building on the work of Potrie and Chung to build a robust class of examples.