Measure preserving maps with bounded total variation

Abstract

Consider a piecewise affine Lipschitz map $\ddot{I}•:\hat{I}\text{c◯}\to \mathbb{R}$, where $\hat{I}\text{c◯}\subset {\mathbb{R}}^d$ is an open set, and assume that $x\mapsto x+t\nabla \ddot{I}•(x)$ is injective for almost every $t>0$. In (J.-G. Liu, R.~L. Pego, \emph{Rigidly breaking potential flows and a countable Alexandrov theorem for polytopes}, Pure Appl. Anal., \textbf{7}(4), 2025) the authors conjecture that every such $\ddot{I}•$ must be locally convex. We prove the result assuming additionally $\nabla \ddot{I}•\in B{V}_{loc}(\hat{I}\text{c◯})$, for a more general class of measure preserving maps.