A spectral study of the Minkowski Curve

Abstract : The so-called Minkowski curve, a fractal meandering one, whose origin seems to go back to Hermann Minkowski, but is nevertheless difficult to trace precisely, appears as an interesting fractal object, and a good candidate for whom aims at catching an overview of spectral properties of fractal curves.\\ The curve is obtained through an iterative process, starting from a straight line which is replaced by eight segments, and, then, repeating this operation. One may note that the length of the curve grows faster than the one of the Koch one. What are the consequences in the case of a diffusion process on this curve ?\\ The topic is of interest. One may also note that, in electromagnetism, fractal antenna, specifically, on the model of the aforementioned curve, which miniaturized design turn out to perfectly fit wideband or broadband transmission, are increasingly used.\\ It thus seemed interesting to us to build a specific Laplacian on those curves. To this purpose, the analytical approach initiated by~J.~Kigami~\cite{Kigami1989},~\cite{Kigami1993}, taken up, developed and popularized by R.~S.~Strichartz~\cite{Strichartz1999}, \cite{StrichartzLivre2006}, appeared as the best suited one. The Laplacian is obtained through a weak formulation, by means of Dirichlet forms, built by induction on a sequence of graphs that converges towards the considered domain. It is these Dirichlet forms that enable one to obtain energy forms on this domain. \\ Laplacians on fractal curves are not that simple to implement. One must of course bear in mind that a fractal curve is topologically equivalent to a line segment. Thus, how can one make a distinction between the spectral properties of a curve, and a line segment ? Dirichlet forms solely depend on the topology of the domain, and not of its geometry. The solution is to consider energy forms more sophisticated than classical ones, by means of normalization constants that could, not only bear the topology, but, also, the geometric characteristics. In the sequel, we give an explicit construction of a Laplacian on the Minkowski curve, with energy forms that bear the geometric characteristic of the structure. The spectrum of the Laplacian is obtained by means of spectral decimation, in the spirit of the works of M. Fukushima and T. Shima . On doing so, we choose three different methods. This enable us to initiate a detailed study of the spectrum.
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Contributeur : Claire David <>
Soumis le : mercredi 20 septembre 2017 - 10:17:42
Dernière modification le : jeudi 11 janvier 2018 - 06:12:16


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  • HAL Id : hal-01527996, version 2



Nizare Riane, Claire David. A spectral study of the Minkowski Curve. 2017. 〈hal-01527996v2〉



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