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 TMF, 1996, Volume 109, Number 3, Pages 323–337 (Mi tmf1230)

Fractal geometry of images of $p$-adic numbers and solenoids continuous immersions to Euclidean spaces

D. V. Chistyakov

Kazan State University

Abstract: Family of continuous maps of $p$-adic numbers $\mathbf Q_p$ and solenoids $\mathbf T_p$ to the complex plane $\mathbf C$ and to the $\mathbf R^3$, respectively, are obtained in an explicit form. Maps for which the Cantor set and the Serpinsky triangle are unitary ball images to $\mathbf Q_2$ and $\mathbf Q_3$, respectively, belong to such families. The subset of immersions for each of that families is found. For these immersions Hausdorff dimensions of images are calculated and it is shown that fractal measure of $\mathbf Q_p$ image coincides with the Haar measure in $\mathbf Q_p$. It is shown, that the image of the $p$-adic solenoid is invariant set with fractal dimension of a some dynamic system. Computer pictures of some fractal images are presented.

DOI: https://doi.org/10.4213/tmf1230

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English version:
Theoretical and Mathematical Physics, 1996, 109:3, 1495–1507

Bibliographic databases:

Citation: D. V. Chistyakov, “Fractal geometry of images of $p$-adic numbers and solenoids continuous immersions to Euclidean spaces”, TMF, 109:3 (1996), 323–337; Theoret. and Math. Phys., 109:3 (1996), 1495–1507

Citation in format AMSBIB
\Bibitem{Chi96} \by D.~V.~Chistyakov \paper Fractal geometry of images of $p$-adic numbers and solenoids continuous immersions to Euclidean spaces \jour TMF \yr 1996 \vol 109 \issue 3 \pages 323--337 \mathnet{http://mi.mathnet.ru/tmf1230} \crossref{https://doi.org/10.4213/tmf1230} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1472471} \zmath{https://zbmath.org/?q=an:0937.28007} \transl \jour Theoret. and Math. Phys. \yr 1996 \vol 109 \issue 3 \pages 1495--1507 \crossref{https://doi.org/10.1007/BF02073866} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1996XN86800001}