Fractal Geometry For Images Of Continuous Map Of p-Adic Numbers And p-Adic Solenoids Into Euclidean Spaces

Abstract

Explicit formulas are obtained for a family of continuous mappings of p-adic numbers $\Qp$ and solenoids $\Tp$ into the complex plane $\sC$ and the space \~$\Rs ^{3}$, respectively. Accordingly, this family includes the mappings for which the Cantor set and the Sierpinski triangle are images of the unit balls in $\Qn{2}$ and $\Qn{3}$. In each of the families, the subset of the embeddings is found. For these embeddings, the Hausdorff dimensions are calculated and it is shown that the fractal measure on the image of $\Qp$ coincides with the Haar measure on $\Qp$. It is proved that under certain conditions, the image of the $p$-adic solenoid is an invariant set of fractional dimension for a dynamic system. Computer drawings of some fractal images are presented.