Geodesic
Fractal geometry of images of $p$-adic numbers and solenoids continuous immersions to Euclidean spaces
Teoretičeskaâ i matematičeskaâ fizika, Tome 109 (1996) no. 3, pp. 323-337 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{TMF_1996_109_3_a0,
     author = {D. V. Chistyakov},
     title = {Fractal geometry of images of $p$-adic numbers and solenoids continuous immersions to {Euclidean} spaces},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {323--337},
     year = {1996},
     volume = {109},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1996_109_3_a0/}
}
TY  - JOUR
AU  - D. V. Chistyakov
TI  - Fractal geometry of images of $p$-adic numbers and solenoids continuous immersions to Euclidean spaces
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 1996
SP  - 323
EP  - 337
VL  - 109
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_1996_109_3_a0/
LA  - ru
ID  - TMF_1996_109_3_a0
ER  - 
%0 Journal Article
%A D. V. Chistyakov
%T Fractal geometry of images of $p$-adic numbers and solenoids continuous immersions to Euclidean spaces
%J Teoretičeskaâ i matematičeskaâ fizika
%D 1996
%P 323-337
%V 109
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_1996_109_3_a0/
%G ru
%F TMF_1996_109_3_a0
D. V. Chistyakov. Fractal geometry of images of $p$-adic numbers and solenoids continuous immersions to Euclidean spaces. Teoretičeskaâ i matematičeskaâ fizika, Tome 109 (1996) no. 3, pp. 323-337. http://geodesic.mathdoc.fr/item/TMF_1996_109_3_a0/

[1] V. S. Vladimirov, I. V. Volovich, E. I. Zelenov, $p$-Adicheskii analiz i matematicheskaya fizika, Nauka, M., 1994 | MR

[2] M. Pitkanen, $p$-adic Physics?, Preprint University of Helsinki, SF-00170, 1994

[3] S. Havlin, N. Weissman, J. Phys. A, 19 (1986), 1021–1026 | DOI | MR

[4] E. I. Zelenov, J. Math. Phys., 32 (1991), 147–152 | DOI | MR | Zbl

[5] A. Likhtenberg, M. Liberman, Regulyarnaya i stokhasticheskaya dinamika, Mir, M., 1984

[6] A. T. Ogielski, D. L. Stein, Phys. Rev. Let., 55:15 (1985) | DOI | MR

[7] K. Allen, M. Kluatr, “Opticheskie preobrazovaniya fraktalov”, Fraktaly v fizike, eds. L. Petronero, E. Tozatti, Mir, M., 1988

[8] G. Federer, Geometricheskaya teoriya mery, Mir, M., 1981 | MR

[9] P. Billingslei, Ergodicheskaya teoriya i informatsiya, Mir, M., 1969 | MR

[10] N. Koblits, $p$-Adicheskie chisla, $p$-adicheskii analiz i dzeta-funktsii, Mir, M., 1982 | MR

[11] E. Khyuit, K. Ross, Abstraktnyi garmonicheskii analiz, T. 1, Nauka, M., 1975

[12] Dzh. Kelli, Obschaya topologiya, Nauka, M., 1981 | MR

[13] I. M. Gelfand, M. I. Graev, L. I. Pyatetskii-Shapiro, Obobschennye funktsii, Vyp. 6, Nauka, M., 1966 | MR

[14] B. V. Shabat, Kompleksnyi analiz, T. 1, Nauka, M., 1985 | MR

[15] E. Feder, Fraktaly, Mir, M., 1991 | MR

[16] Dzh. Zaiman, Modeli besporyadka, Mir, M., 1982

[17] V. I. Arnold, Obyknovennye differentsialnye uravneniya, Nauka, M., 1975 | MR

[18] D. Mamford, Lektsii o teta-funktsiyakh, Mir, M., 1988 | MR