Linear dilatation and differentiability of homeomorphisms of $\mathbb {R}^n$

Abstract:

According to a classical result, if $\Omega$ is a domain in $\mathbb {R}^d$, where $d>1$, $f: \Omega \rightarrow \mathbb {R}^d$ is a homeomorphism and the lim-sup dilatation $H_f$ of $f$ is finite almost everywhere on $\Omega$, then $f$ is differentiable almost everywhere on $\Omega$. We show that this theorem fails if $H_f$ is replaced by the lim-inf dilatation $h_f$. Our example demonstrates the sharpness of recent results of Kallunki and Koskela concerning the $h_f$ function and also of Balogh and Csörnyei involving the lower-scaled oscillation of continuous functions $f: \Omega \rightarrow \mathbb {R}$.