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Journal Articles Contemporary mathematics Year : 2023

Iterated Fractal Drums ∼ Some New Perspectives: Polyhedral Measures, Atomic Decompositions and Morse Theory

Claire David
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Abstract

In this research paper, we carry on our exploration of the connections between the Complex Fractal Dimensions of an iterated fractal drum (IFD) and the intrinsic properties of the fractal involved – in our present case, the Weierstrass Curve. In order to gain a better understanding of the differential operators associated to this everywhere singular object, we use atomic decompositions, which enable us to characterize Besov spaces on the Weierstrass Curve ΓW and then, by means of trace theorems which extend the results of Alf Jonsson and Hans Wallin obtained in the case of a d-set, to identify the trace of the classical Sobolev spaces on Γw. For this purpose, we construct a specific polyhedral measure, which is done by means of a sequence of polygonal neighborhoods of the Curve. We then determine the order of the fractal Laplacian on Γw, thanks to the connections between Sobolev spaces and the usual Laplacian. Moreover, we lay out some of the foundations of an extension of Morse theory dedicated to fractals, where the Complex Fractal Dimensions appear to play a major role, by means of level sets connected to the successive prefractal approximations. In the end, we envision the Weierstrass Curve as the projection of a 3-dimensional vertical comb, where each horizontal row is associated to the kth cohomogical infinitesimal, the fractal signature of the kth prefractal approximation, according to our previous results on fractal cohomology.
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Dates and versions

hal-03946104 , version 1 (19-01-2023)
hal-03946104 , version 2 (04-06-2023)
hal-03946104 , version 3 (10-01-2024)

Identifiers

  • HAL Id : hal-03946104 , version 3

Cite

Claire David, Michel L Lapidus. Iterated Fractal Drums ∼ Some New Perspectives: Polyhedral Measures, Atomic Decompositions and Morse Theory. Contemporary mathematics, In press. ⟨hal-03946104v3⟩
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