Fractal geometry for images of continuous embeddings of \(p\)-adic numbers and solenoids into Euclidean spaces. (English. Russian original) Zbl 0937.28007
Theor. Math. Phys. 109, No. 3, 1495-1507 (1996); translation from Teor. Mat. Fiz. 109, No. 3, 323-337 (1996).
Summary: Explicit formulas are obtained for a family of continuous mappings of \(p\)-adic numbers \(\mathbb{Q}_p\) and solenoids \(\mathbb{T}_p\) into the complex plane \(\mathbb{C}\) and the space \(\mathbb{R}^3\), respectively. Accordingly, this family includes the mappings for which the Cantor set and the Sierpiński carpet are images of the unit balls in \(\mathbb{Q}_2\) and \(\mathbb{Q}_3\). 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 \(\mathbb{Q}_p\) coincides with the Haar measure on \(\mathbb{Q}_p\). It is proved that under certain conditions, the image of the \(p\)-adic solenoid is an invariant set of fractional dimension for a dynamical system. Computer drawings of some fractal images are presented.
MSC:
| 28A80 | Fractals |
| 11K41 | Continuous, \(p\)-adic and abstract analogues |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 11S80 | Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) |
Keywords:
\(p\)-adic numbers; solenoids; Cantor set; Sierpiński carpet; Hausdorff dimensions; fractal measure; Haar measure; fractional dimensionReferences:
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