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

Sets of non-differentiability for conjugacies between expanding interval maps