Calculus of Variations and Geometric Measure Theory
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E. Durand-Cartagena - J. Gong - J. Á. Jaramillo

Sierpinski-type fractals are differentiably trivial

created by durandcar on 13 Jul 2018

[BibTeX]

Preprint

Inserted: 13 jul 2018
Last Updated: 13 jul 2018

Year: 2018

Abstract:

In this note we investigate the viability of generalized Rademacher theorems on a certain class of fractals in Euclidean spaces. Such sets are not necessarily self-similar, but satisfy a weaker "scale-similar" property; in particular, they include the non self-similar carpets introduced by Mackay-Tyson-Wildrick but with diff erent scale ratios. Speci fically we identify certain geometric properties enjoyed by these fractals and, in the case that they have zero Lebesgue measure, we show that such fractals cannot support nonzero derivations in the sense of Weaver. As a result such fractals cannot be Lipschitz di erentiability spaces in the sense of Cheeger and Keith.


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