×

zbMATH — the first resource for mathematics

On a generalized self-similarity in the \(p\)-adic field. (English) Zbl 1357.28012

MSC:
28A80 Fractals
11S82 Non-Archimedean dynamical systems
37P25 Dynamical systems over finite ground fields
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] 1. J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J.30 (1981) 713-747. genRefLink(16, ’S0218348X16500419BIB001’, ’10.1512%252Fiumj.1981.30.30055’); genRefLink(128, ’S0218348X16500419BIB001’, ’A1981ML89900006’);
[2] 2. K. J. Falconer, Fractal Geometry (John Wiley & Sons, Chinchester, 1990).
[3] 3. B. B. Mandelbrot, Intermitten turbulence in self-similar cascades: Divergence of high moments and dimension of the carrier, J. Fluid Mech.62 (1974) 331-358. genRefLink(16, ’S0218348X16500419BIB003’, ’10.1017%252FS0022112074000711’); genRefLink(128, ’S0218348X16500419BIB003’, ’A1974S274000008’);
[4] 4. M. Barnsley, Fractals Everuwhere (Academic Press, Boston, 1988).
[5] 5. T. Kamae, J. Luo and B. Tan, A gluing lemma for iterated function systems, Fractals23 (2015) 155019. [Abstract] genRefLink(128, ’S0218348X16500419BIB005’, ’000355379400019’);
[6] 6. M. Saltan and B. Demir, An iterated function system for the closure of the adding machine group, Fractals23 (2015) 155033. [Abstract] genRefLink(128, ’S0218348X16500419BIB006’, ’000358787500012’);
[7] 7. J. Vass, On intersecting IFS fractals with lines, Fractals22 (2014) 1450014. [Abstract] genRefLink(128, ’S0218348X16500419BIB007’, ’000345091000008’);
[8] 8. A. F. Beardon and Ch. Pommerenke, The Poincare metric of plane domains, J. Lond. Math. Soc.18 (1978) 475-483. genRefLink(16, ’S0218348X16500419BIB008’, ’10.1112%252Fjlms%252Fs2-18.3.475’); genRefLink(128, ’S0218348X16500419BIB008’, ’A1978GH94700012’);
[9] 9. M. L. Blank, Stability and Localization in Chatic Dynamics (MTsNMO, Moscow, 2001).
[10] 10. D. R. Guy and S. Stephen, Fractured fractals and broken dreams: Self-similar geometry through metric and measure, in Oxford Lecture Ser. Math. Appl., Vol. 7 (Oxford Science Publications, Oxford University Press, UK, 1997). · Zbl 0887.54001
[11] 11. P. Jarvi and M. Vuorinen, Uniformly perfect sets and quasiregular mappings, J. Lond. Math. Soc.54 (1996) 515-529. genRefLink(16, ’S0218348X16500419BIB011’, ’10.1112%252Fjlms%252F54.3.515’); genRefLink(128, ’S0218348X16500419BIB011’, ’A1996VZ30900007’);
[12] 12. V. S. Anashin and A. Khrennikov, Applied Algebraic Dynamics (Walter de Gruter Co., Berlin, 2009). genRefLink(16, ’S0218348X16500419BIB012’, ’10.1515%252F9783110203011’); · Zbl 1184.37002
[13] 13. R. L. Benedetto, Hyperbolic maps in p-adic dynamics, Erg. Theor. Dyn. Sys.21 (2001) 1-11. genRefLink(128, ’S0218348X16500419BIB013’, ’000167424600001’); · Zbl 0972.37027
[14] 14. R. L. Benedetto, Reduction, dynamics and Julia sets of rational functions, J. Numb. Theory86 (2001) 175-195. genRefLink(16, ’S0218348X16500419BIB014’, ’10.1006%252Fjnth.2000.2577’); genRefLink(128, ’S0218348X16500419BIB014’, ’000167222600001’);
[15] 15. A. H. Fan, M. T. Li, J. Y. Yao and D. Zhou, Strict ergodicity of affine p-adic dynamical systems on \(\mathbb{Z}\)p, Adv. Math.214 (2007) 666-700. genRefLink(16, ’S0218348X16500419BIB015’, ’10.1016%252Fj.aim.2007.03.003’); genRefLink(128, ’S0218348X16500419BIB015’, ’000249626100007’);
[16] 16. A. H. Fan, L. M. Liao, Y. F. Wang and D. Zhou, p-adic repellers in \(\mathbb{Q}\)p are subshifts of finite type, C. R. Math. Acad. Sci. Paris344 (2007) 219-224. genRefLink(16, ’S0218348X16500419BIB016’, ’10.1016%252Fj.crma.2006.12.007’); genRefLink(128, ’S0218348X16500419BIB016’, ’000244924000002’); · Zbl 1108.37016
[17] 17. V. M. Gundlach, A. Khrennikov and K. O. Lindahl, On ergodic behavior of p-adic dynamical systems. Infin. Dimen. Anal. Quantum Probab. Relat. Top.4 (2001) 569-577. [Abstract] genRefLink(128, ’S0218348X16500419BIB017’, ’000173461100009’); · Zbl 1040.37005
[18] 18. M. Khamraev and F. Mukhamedov, On a class of rational p-adic dynamical systems, J. Math. Anal. Appl.315 (2006) 76-89. genRefLink(16, ’S0218348X16500419BIB018’, ’10.1016%252Fj.jmaa.2005.08.041’); genRefLink(128, ’S0218348X16500419BIB018’, ’000235104500007’); · Zbl 1085.37040
[19] 19. A. Yu. Khrennikov and M. Nilsson, p-Adic Deterministic and Random Dynamical Systems (Kluwer, Dordreht, 2004). genRefLink(16, ’S0218348X16500419BIB019’, ’10.1007%252F978-1-4020-2660-7’);
[20] 20. U. A. Rozikov and I. A. Sattarov, On a nonlinear p-adic dynamical systems, p-adic Numbers, Ultrametric Anal. Appl.6 (2014) 54-65. genRefLink(16, ’S0218348X16500419BIB020’, ’10.1134%252FS207004661401004X’); · Zbl 1347.37148
[21] 21. C. F. Woodcock and N. P. Smart, p-adic chaos and random number generation, Exp. Math.7 (1998) 333-342. genRefLink(16, ’S0218348X16500419BIB021’, ’10.1080%252F10586458.1998.10504379’); genRefLink(128, ’S0218348X16500419BIB021’, ’000078254300005’);
[22] 22. W. Y. Qiu, Y. F. Wang, J. H. Yang and Y. C. Yin, On metric properties of limiting sets of contractive analytic non-Archimedean dynamical systems. J. Math. Anal. App.414 (2014) 386-401. genRefLink(16, ’S0218348X16500419BIB022’, ’10.1016%252Fj.jmaa.2014.01.015’); genRefLink(128, ’S0218348X16500419BIB022’, ’000332194700029’); · Zbl 1391.37088
[23] 23. J. H. Silverman, The Arithmetic of Dynamical Systems, Graduate Texts in Mathematics 241 (Springer, New York, 2007). genRefLink(16, ’S0218348X16500419BIB023’, ’10.1007%252F978-0-387-69904-2’);
[24] 24. F. Mukhamedov, Recurrence equations over trees in a non-Archimedean context, P-adic Numb. Ultra. Anal. Appl.6 (2014) 310-317. genRefLink(16, ’S0218348X16500419BIB024’, ’10.1134%252FS2070046614040062’); · Zbl 1420.47027
[25] 25. F. Mukhamedov and H. Akin, On non-Archimedean recurrence equations and their applications, J. Math. Anal. Appl.423 (2015) 1203-1218. genRefLink(16, ’S0218348X16500419BIB025’, ’10.1016%252Fj.jmaa.2014.10.046’); genRefLink(128, ’S0218348X16500419BIB025’, ’000359756100019’); · Zbl 1303.39011
[26] 26. F. Mukhamedov, On existence of generalized Gibbs measures for one dimensional p-adic countable state Potts model, Proc. Steklov Inst. Math.265 (2009) 165-176. genRefLink(16, ’S0218348X16500419BIB026’, ’10.1134%252FS0081543809020163’); genRefLink(128, ’S0218348X16500419BIB026’, ’000268514300016’);
[27] 27. D. Gandolfo, U. Rozikov and J. Ruiz, On p-adic Gibbs measures for hard core model on a Cayley Tree, Markov Proc. Rel. Topics18 (2012) 701-720. genRefLink(128, ’S0218348X16500419BIB027’, ’000313221100008’); · Zbl 1281.82006
[28] 28. F. Mukhamedov, A dynamical system approach to phase transitions for p-adic Potts model on the Cayley tree of order two, Rep. Math. Phys.70 (2012) 385-406. genRefLink(16, ’S0218348X16500419BIB028’, ’10.1016%252FS0034-4877%252812%252960053-6’); genRefLink(128, ’S0218348X16500419BIB028’, ’000313085600009’);
[29] 29. F. Mukhamedov, On dynamical systems and phase transitions for q+1-state p-adic Potts model on the Cayley tree, Math. Phys. Anal. Geom.16 (2013) 49-87. genRefLink(16, ’S0218348X16500419BIB029’, ’10.1007%252Fs11040-012-9120-z’); genRefLink(128, ’S0218348X16500419BIB029’, ’000316020400003’);
[30] 30. F. Mukhamedov, On strong phase transition for one dimensional countable state p-adic Potts model, J. Stat. Mech. (2014) P01007. genRefLink(128, ’S0218348X16500419BIB030’, ’000332089600007’); · Zbl 1331.82016
[31] 31. F. Mukhamedov, Renormalization method in p-adic \(\lambda\)-model on the Cayley tree, Int. J. Theor. Phys.54 (2015) 3577-3595. genRefLink(16, ’S0218348X16500419BIB031’, ’10.1007%252Fs10773-015-2597-z’); genRefLink(128, ’S0218348X16500419BIB031’, ’000361005700010’);
[32] 32. F. Mukhamedov and O. Khakimov, Phase transition and chaos: p-adic Potts model on a Cayley tree, Chaos Solitons Fractals87 (2016) 190-196. genRefLink(16, ’S0218348X16500419BIB032’, ’10.1016%252Fj.chaos.2016.04.003’); genRefLink(128, ’S0218348X16500419BIB032’, ’000377229500022’);
[33] 33. A. M. Robert, A Course of p-Adic Analysis (Springer, New York, 2000). genRefLink(16, ’S0218348X16500419BIB033’, ’10.1007%252F978-1-4757-3254-2’); · Zbl 0947.11035
[34] 34. N. Koblitz, p-Adic Numbers, p-Adic Analysis and Zeta-Function (Springer, Berlin, 1977). genRefLink(16, ’S0218348X16500419BIB034’, ’10.1007%252F978-1-4684-0047-2’); · Zbl 0364.12015
[35] 35. A. Khrennikov, F. Mukhamedov and J. F. F. Mendes, On p-adic Gibbs measures of countable state Potts model on the Cayley tree, Nonlinearity20 (2007) 2923-2937. genRefLink(16, ’S0218348X16500419BIB035’, ’10.1088%252F0951-7715%252F20%252F12%252F010’); genRefLink(128, ’S0218348X16500419BIB035’, ’000252477600010’);
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.