Sets of non-differentiability for conjugacies between expanding interval maps

TL;DR

This paper investigates the differentiability properties of conjugacies between expanding interval maps, revealing a multifractal structure in the set where derivatives do not exist, and establishing a rigidity dichotomy based on Hausdorff dimension.

Contribution

It provides an explicit Hausdorff dimension formula for non-differentiability sets using multifractal analysis and thermodynamic formalism, and introduces a rigidity dichotomy for conjugacies.

Findings

Hausdorff dimension of non-differentiability set is between zero and one

Dimension is explicitly determined by the Lyapunov spectrum

Results imply a rigidity dichotomy for conjugacies

Abstract

We study differentiability of topological conjugacies between expanding piecewise $C^{1+\epsilon}$ interval maps. If these conjugacies are not $C^1$, then they have zero derivative almost everywhere. We obtain the result that in this case the Hausdorff dimension of the set of points for which the derivative of the conjugacy does not exist lies strictly between zero and one. Using multifractal analysis and thermodynamic formalism, we show that this Hausdorff dimension is explicitly determined by the Lyapunov spectrum. Moreover, we show that these results give rise to a "rigidity dichotomy" for the type of conjugacies under consideration.