New Insights for Fractal Zeta Functions Polyhedral Neighborhoods vs Tubular Neighborhoods - Sorbonne Université Access content directly
Preprints, Working Papers, ... Year : 2023

New Insights for Fractal Zeta Functions Polyhedral Neighborhoods vs Tubular Neighborhoods

Claire David
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  • PersonId : 1262788
  • IdHAL : cldavid
Michel L Lapidus
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  • PersonId : 883127

Abstract

In this note, which is an announcement of the long paper [8], we introduce new fractal zeta functions, associated with polyhedral neighborhoods, better suited to fractals when exact expressions for the volume of tubular neighborhoods cannot be computed. Accordingly, in the model-and very significant-case of the Weierstrass Curve Γ W , we give exact expressions for the volume of polyhedral neighborhoods for the sequence of prefractal graphs which converge to Γ W-the so-called Weierstrass Iterated Fractal Drums (in short, Weierstrass IFDs). Those IFDs are associated with a suitable (and geometrically meaningful) sequence of small parameters tending to zero, also known as the cohomology infinitesimals, due to their connections with fractal cohomology. We also introduce the associated local and global polyhedral fractal zeta functions. The local fractal zeta functions consist in the sequence of zeta functions associated with the sequence of polyhedral neighborhoods, and satisfy a recurrence relation, which enables us to prove that the poles of the limit fractal zeta function-the global zeta function, associated with the limit fractal object-are exactly the same as the Complex Dimensions of the Weierstrass function itself. This result makes the connection with fractal cohomology, where, for any nonnegative integer m, the m t h cohomology group is comprised of continuous functions which possess a generalized Taylor expansion, with fractional derivatives of orders the underlying-and actual-Complex Dimensions. By using the aforementioned exact expressions of the polyhedral neighborhoods, we also revisit the computation of the box-counting (or Minkowski) dimension of the Weierstrass Curve, in a fully rigorous manner and therefore prove part of Mandelbrot's conjecture concerning the fractal dimension of the Weierstrass Curve.
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Dates and versions

hal-04306826 , version 1 (25-11-2023)

Identifiers

  • HAL Id : hal-04306826 , version 1

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Claire David, Michel L Lapidus. New Insights for Fractal Zeta Functions Polyhedral Neighborhoods vs Tubular Neighborhoods. 2023. ⟨hal-04306826⟩
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