Sets of nondifferentiability for conjugacies between expanding interval maps
Volume 206 / 2009
Fundamenta Mathematicae 206 (2009), 161-183
MSC: Primary 37C45; Secondary 28A80, 37A10.
DOI: 10.4064/fm206-0-10
Abstract
We study differentiability of topological conjugacies between expanding piecewise $C^{1+\epsilon }$ interval maps. If these conjugacies are not $C^1$, then their derivative vanishes Lebesgue almost everywhere. We show 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. Moreover, by employing the thermodynamic formalism, we show that this Hausdorff dimension can be determined explicitly in terms of the Lyapunov spectrum. These results then give rise to a “rigidity dichotomy” for the type of conjugacies under consideration.
