Abstract
Explicit formulas are obtained for a family of continuous mappings of p-adic numbersQ p and solenoidsT p into the complex planeC and the spaceR 3, respectively. Accordingly, this family includes the mappings for which the Cantor set and the Sierpiński carpet are images of the unit balls inQ 2 andQ 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 ofQ p coincides with the Haar measure onQ p. 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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By a dynamic system in Rn we mean an autonomous system ofn first-order equations that satisfies the conditions of the existence and uniqueness theorem.
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No. 3, pp. 323–337, December, 1996.
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Chistyakov, D.V. Fractal geometry for images of continuous embeddings ofp-adic numbers and solenoids into Euclidean spaces. Theor Math Phys 109, 1495–1507 (1996). https://doi.org/10.1007/BF02073866
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DOI: https://doi.org/10.1007/BF02073866