Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 635-655

SIERPINSKI-TYPE FRACTALS ARE DIFFERENTIABLY TRIVIAL

Estibalitz Durand-Cartagena, Jasun Gong and Jesús A. Jaramillo

UNED, ETSI Industriales, Departamento de Matemática Aplicada
28040 Madrid, Spain; edurand 'at' ind.uned.es

Fordham University, Department of Mathematics
441 E. Fordham Rd, Bronx, NY 10458, United States; jgong7 'at' fordham.edu

Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, IMI and Departamento de Análisis Matemático y Matemática Aplicada
28040 Madrid, Spain; jaram 'at' mat.ucm.es

Abstract. In this note we study generalized differentiability of functions on a 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 [12] but with different scale ratios. Specifically we identify certain geometric criteria for 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 [16]. As a result (Theorem 26) such fractals cannot support Alberti representations and in particular, they cannot be Lipschitz differentiability spaces in the sense of Cheeger [3] and Keith [9].

2010 Mathematics Subject Classification: 31E05, 28A80, 13N15.

Key words: Sierpinski-type fractals, doubling measure, metric derivations.

Reference to this article: E. Durand-Cartagena, J. Gong and J. A. Jaramillo: Sierpinski-type fractals are differentiably trivial. Ann. Acad. Sci. Fenn. Math. 44 (2019), 635-655.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4460