Theoretical and Mathematical Physics

, Volume 109, Issue 3, pp 1495–1507 | Cite as

Fractal geometry for images of continuous embeddings ofp-adic numbers and solenoids into Euclidean spaces

  • D. V. Chistyakov


Explicit formulas are obtained for a family of continuous mappings of p-adic numbersQp and solenoidsTp into the complex planeC and the spaceR3, respectively. Accordingly, this family includes the mappings for which the Cantor set and the Sierpiński carpet are images of the unit balls inQ2 andQ3. 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 ofQp coincides with the Haar measure onQp. 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.


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Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • D. V. Chistyakov
    • 1
  1. 1.Kazan' State UniversityUSSR

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