We study differentiability of topological conjugacies between expanding piecewise interval maps. If these conjugacies are not C¹, 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm206-0-10,
author = {Thomas Jordan and Marc Kesseb\"ohmer and Mark Pollicott and Bernd O. Stratmann},
title = {Sets of nondifferentiability for conjugacies between expanding interval maps},
journal = {Fundamenta Mathematicae},
volume = {205},
year = {2009},
pages = {161-183},
zbl = {1187.37035},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm206-0-10}
}
Thomas Jordan; Marc Kesseböhmer; Mark Pollicott; Bernd O. Stratmann. Sets of nondifferentiability for conjugacies between expanding interval maps. Fundamenta Mathematicae, Tome 205 (2009) pp. 161-183. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm206-0-10/