Extreme non-differentiability of typical Lipschitz mappings
Michael Dymond, Olga Maleva
TL;DR
The paper addresses how typical Lipschitz mappings differentiate on sets in normed spaces, showing that generically a $1$-Lipschitz map is extremely non-differentiable: at typical points of a set $E$ the derivative ratios approximate every operator in a prescribed unit-ball subspace. The authors employ Banach-Mazur game arguments and a detailed construction to prove two main results: (i) a residual set of mappings exhibits derivative behavior containing the unit ball of any separable subspace $W$ of $\mathcal{L}(X,Y)$, and (ii) when $X$ is finite-dimensional and $E$ is $F_{σ}$ purely unrectifiable, this holds at every point of $E$. A smooth approximation framework is developed to support the arguments, enabling controlled perturbations and local-to-global Lipschitz estimates. These results extend and sharpen existing scalar-valued results to vector-valued Lipschitz maps with arbitrary norms, highlighting the ubiquity of extreme nondifferentiability in a broad setting.
Abstract
We show that no matter what subset of a normed space is given, a typical 1-Lipschitz mapping into a Banach space is non-differentiable at a typical point of the set in a very strong sense: the derivative ratio approximates, on arbitrary small scales, every linear operator of norm at most 1. For subsets of finite-dimensional normed spaces which can be covered by a countable union of closed purely unrectifiable sets this extreme non-differentiability holds for a typical Lipschitz mapping at every point. Both results are new even for Lipschitz mappings with a finite-dimensional co-domain.