  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.
