On the approximate differentiability of inverse maps

Abstract

A chain rule for an a.e. approximately differentiable homeomorphism ${{f: \Omega \subset \mathbb{R}^{n} \xrightarrow{\rm onto}} \Omega^{\prime} \subset \mathbb{R}^{n}}$ is proved. Namely, there exists a Borel set ${{B \subset \mathcal{R}_{f}}}$ , where

$$\mathcal{R}_{f} = \{x \in \Omega : f\, {\rm is\,approximately\,differentiable\,at}\,x\,{\rm and}\,J_{f}(x) \neq 0\},$$

such that ${{\mid \mathcal{R}_{f} \setminus B \mid = 0}}$ , ${{f(B) \subset \mathcal{R}_{f-1}}}$ and J _f-1(f(x))J _f (x) =  1 for all ${{x \in B}}$ . In general, it is not true that ${{f(\mathcal{R}_{f}) \subset \mathcal{R}_{f-1}}}$ . Indeed, there exists a homeomorphism ${f_{0}: \Omega \subset \mathbb{R}^{n} \xrightarrow {\rm onto} \Omega^{\prime} \subset \mathbb{R}^{n}}$ such that f ₀ is approximately differentiable at x ₀ with ${{J_{f0}(x_{0}) \neq 0}}$ and ${{{f_{0}^{-1}}}}$ is not approximately differentiable at f ₀(x ₀). We give various conditions to guarantee that a homeomorphism preserves density points.