p-adic dynamics. (English) Zbl 0672.58019

The quadratic map \(x\to f(x)=x^ 2+a\) is studied over p-adic numbers. It is shown that p-adic dynamics exhibits many similarities with real or complex dynamics. In particular, the notion of attractive or indifferent cycles, quasiperiodicity and chaos are still applicable. On the other hand in the case of p-adic dynamics chaos does not emerge through a cascade of period-doubling bifurcations. The procedure for finding all attractive cycles is given.
The quadratic map is then studied in some detail near an indifferent fixed point by showing that, under certain conditions, it is topologically conjugated to a linear map. The analysis here is much simpler than in the complex case. It is proven that for \(| a|_ p>1\) most points of \(Q_ p\) eventually end up at infinity under iterations of f; points with a bounded orbit exist only if \(a=-\gamma^ 2\). They belong to the Cantor set \(\Lambda\) on which f takes a form of shift map. Due to the simplicity of this map it can be completely understood. In this way it is proven that the p-adic quadratic map with \(| a|_ p>1\) is chaotic on the Cantor set \(\Lambda\).
Reviewer: P.Maslanka


37B99 Topological dynamics
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
43A99 Abstract harmonic analysis
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[1] R. L. Devaney,An Introduction to Chaotic Dynamical Systems (Benjamin/Cummings, 1986); H. O. Peitgen and P. H. Richter,The Beauty of Fractals (Springer-Verlag, Berlin, 1986).
[2] P. Cvitanovic,Universality in Chaos (Adam Hilger, Bristol, 1984), and references therein.
[3] N. Koblitz,p-adic Numbers, p-adic Analysis and Zeta Functions (Springer-Verlag, Berlin, 1984); K. Malher,p-adic Numbers and their Functions (Cambridge University Press, Cambridge, 1983). · Zbl 0494.12009
[4] I. M. Gel’fand, M. I. Graev, and I. I. Pyatetskii-Shapiro,Representation Theory and Automorphic Functions (Saunders, London, 1966).
[5] I. V. Volovich,Classical Quantum Gravity 4:L83 (1987); B. Grossmann, Rockefeller University Preprint DOE/ER/40325-7-TASK B (1987); P. G. O. Freund and M. Olson,Nucl. Phys. B 297:86 (1988); Y. Meurice, Argonne National Laboratory Preprint ANL-HEP-PR-87-114 (1987); G. Parisi,Mod. Phys. Lett. A 1988:639; V. S. Vladimirov and I. V. Volovich, Preprint SMI O1/88 (1988); C. Alacoqueet al., UCL-IPT-88-05,Phys. Lett. B 211:59 (1988). · doi:10.1088/0264-9381/4/4/003
[6] P. G. O. Freund and M. Olson,Phys. Lettt. B 199:186 (1987). · doi:10.1016/0370-2693(87)91356-6
[7] P. G. O. Freund and E. Witten,Phys. Lett. B 199:191 (1987). · doi:10.1016/0370-2693(87)91357-8
[8] J. L. Gervais,Phys. Lett. B 201:306 (1988); E. Marinari and G. Parisi,Phys. Lett. B 203:52 (1988); H. Yamakoshi,Phys. Lett. B 207:426 (1988); I. Ya. Arefeva, B. G. Dragovic, and I. V. Volovich, Preprint IF-14/88 (1988); L. Brekke, P. G. O. Freund, M. Olson, and E. Witten,Nucl. Phys. B 302:365 (1988). · doi:10.1016/0370-2693(88)91145-8
[9] J. H. Hannay and M. V. Berry,Physica 1D:267 (1980); Y. Nambu, Field theory of Galois’fields, in E. S. Fradkin Festschrift.
[10] C. N. Yang, inSchr?dinger, Centenary Celebration of a Polymath (Cambridge University Press, Cambridge, 1987).
[11] R. Rammal, G. Toulouse, and M. A. Virasoro,Rev. Mod. Phys. 58:765 (1986); B. Grossmann, Rockefeller University Preprint DOE/ER/40325-8-TASKB (1987). · doi:10.1103/RevModPhys.58.765
[12] S. Ben-Menahem,p-adic Iterations, Preprint TAUP 1627-88 (1988).
[13] C. L. Siegel and J. Moser,Lectures on Celestial Mechanics (Springer-Verlag, New York, 1971). · Zbl 0312.70017
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