A remark on uniform expansion

TL;DR

The paper demonstrates that within any neighborhood of volume-preserving smooth diffeomorphisms on the 2-torus, there exists a finitely supported measure that exhibits uniform expansion.

Contribution

It establishes the existence of finitely supported uniformly expanding measures in any neighborhood of volume-preserving diffeomorphisms on the torus.

Findings

Existence of finitely supported uniformly expanding measures

Applicable to any neighborhood in the space of volume-preserving diffeomorphisms

Advances understanding of expansion properties in dynamical systems

Abstract

For every $\mathcal{U} \subset \mathrm{Diff}^\infty_{vol}(\mathbb{T}^2)$ there is a measure of finite support contained in $\mathcal{U}$ which is uniformly expanding.