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Weierstrass Fractal Drums -I − A Glimpse of Complex Dimensions

Abstract : We establish a fractal tube formula for the Weierstrass Curve, which gives, for small values of a strictly positive parameter ε, an explicit expression for the volume of the ε-neighborhood of the Curve. For this purpose, we prove new geometric properties of the Curve and of the associated function, in relation with its local Hölder and reverse Hölder continuity, with explicit estimates that had not been obtained before. We also show that the Codimension 2 − D W is the optimal Hölder exponent for the Weierstrass function W , from which it follows that, as is well known, W is nowhere differentiable. Then, the formula, that yields the expression of the ε-neighborhood, consists of a fractal power series, with underlying exponents the Complex Codimensions. This enables us to obtain the associated tube and distance fractal zeta functions, whose poles yield the set of Complex Dimensions. We prove that the nonzero Complex Dimensions are periodically distributed along countably many vertical lines, with the same oscillatory period. By considering the lower Minkowski content of the Curve, which we prove to be strictly positive, we then show that the Weierstrass Curve is Minkowski nondegenerate, as well as not Minkowski measurable, but admits a nontrivial average Minkowski content-and that, as expected, the Minkowski dimension (or box dimension) D W is the Complex Dimension with maximal real part, and zero imaginary part.
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https://hal.sorbonne-universite.fr/hal-03642326
Contributor : Claire David Connect in order to contact the contributor
Submitted on : Friday, April 15, 2022 - 3:25:29 AM
Last modification on : Thursday, April 21, 2022 - 3:34:32 AM

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  • HAL Id : hal-03642326, version 1

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Claire David, Michel Lapidus. Weierstrass Fractal Drums -I − A Glimpse of Complex Dimensions. 2022. ⟨hal-03642326⟩

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