Differential and Integral Operations in Hidden Spherical Symmetry and the Dimension of the Koch Curve

L. N. Lyakhov ; E. Sanina

Matematičeskie zametki, Tome 113 (2023) no. 4, pp. 517-528

Résumé

Examples of differential and integral operations are given whose dimension is modified as a result of the introduction of new radial variables. Based on the integral measure $x^\gamma\,dx$, $\gamma>-1$, with a weak singularity, we introduce an operator that is interpreted as the Laplace operator in the space of functions of a fractional number of variables. The integration with respect to the measure $x^\gamma\,dx$, $\gamma>-1$, can also be interpreted as the integration over a domain of fractional dimension. The coefficient $\gamma>-1$ of hidden spherical symmetry is introduced. A formula is obtained that relates this coefficient to the Hausdorff dimension of a set in $\mathbb{R}_n$ and the Euclidean dimension $n$. The existence of hidden spherical symmetries is verified by calculating the dimension of the $m$th generation of the Koch curve for arbitrary positive integer $m$.

Détail

Keywords: Laplace operator, Kipriyanov operator, Laplace–Bessel–Kipriyanov operator, singular differential Bessel operator, fractal, self-similarity, integral measure, Hausdorff–Besikovich dimension, Koch curve, generations of the Koch curve.
Mots-clés : fractional dimension, Hausdorff dimension, fractal dimension