1996
DOI: 10.1007/bf02073866 View full text |Buy / Rent full text
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Abstract: Explicit formulas are obtained for a family of continuous mappings of p-adic numbers Q p and solenoids T p into the complex plane C and the space R 3 , respectively. Accordingly, this family includes the mappings for which the Cantor set and the Sierpinski triangle are images of the unit balls in Q 2 and Q 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 Q p coincides with t… Show more

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“…One may wish to somehow visualize their action on Z 2 . Recall that the space Z 2 is not Euclidean and totally disconnected, which makes it difficult to represent graphically [8,11]. It is known to be homeomorphic to the Cantor ternary set, which has Lebesgue measure 0.…”
Section: Cyclesmentioning
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“…One may wish to somehow visualize their action on Z 2 . Recall that the space Z 2 is not Euclidean and totally disconnected, which makes it difficult to represent graphically [8,11]. It is known to be homeomorphic to the Cantor ternary set, which has Lebesgue measure 0.…”
Section: Cyclesmentioning
“…Another kind of description of the Haar measure on the p-adic integers can also be found in Hewitt and Ross [8, p. 220]. One can find a construction of the Haar measure on the p-adic solenoid in Chistyakov [6,Section 3]. It is based on Hausdorff measures and rather sophisticated, while our simpler construction (Theorem 5.1) is based on a probabilistic method and reflects the structure of the p-adic solenoid.…”
Section: If the Row Sumsmentioning
“…In recent years, p-adic analysis has been used in various areas of mathematics as well as in aspects of quantum physics and string theory (Lapidus and van Frankenhuijsen 2006). For a detailed analysis of fractal string and p-adic integers, one may refer to (Chistyakov 1996; Hung 2007; Koblitz 1984; Robert 2000; Schikhof 1984; Vladimirov et al 1994). …”
Section: Introductionmentioning