Stochastic and deterministic characteristics of orbits in chaotically looking dynamical systems

Arnold, V.

doi:10.1090/S0077-1554-09-00180-0

Stochastic and deterministic characteristics of orbits in chaotically looking dynamical systems
HTML articles powered by AMS MathViewer

by V. I. Arnold
Translated by: O. Khleborodova

Trans. Moscow Math. Soc. 2009, 31-69

DOI: https://doi.org/10.1090/S0077-1554-09-00180-0

Published electronically: December 3, 2009

PDF | Request permission

Abstract:

We study finite length sequences of numbers which, at the first glance, look like realizations of a random variable (for example, sequences of fractional parts of arithmetic and geometric progressions, last digits of sequences of prime numbers, and incomplete periodic continuous fractions).

The degree of randomness of a finite length sequence is measured by the parameter introduced by Kolmogorov in his 1933 Italian article published in an actuarial journal.

Unexpectedly, fractional parts of terms of a geometric progression behave much more randomly than terms of an arithmetic progression, and the statistics of periods of continuous fractions for eigenvalues of unimodular matrices turns out to be different from the classical Gauss–Kuzmin statistics of partial continuous fractions of random real numbers.

Empirically, the lengths of the period of continuous fractions for the roots of quadratic equations with leading coefficient 1 and increasing other (integer) coefficients, grow, on the average, as the square root of the discriminant of the equation.

References

A. Kolmogoroff, Sulla determinazione empirica di una legge di distribuzione, Giorn. Ist. Ital. Attuari. 4 (1933), 83–91.
A. N. Kolmogoroff, On a new confirmation of Mendel’s laws, C. R. (Doklady) Acad. Sci. URSS (N.S.) 27 (1940), 37–41. MR 0003556
V. I. Arnold, Dynamics, statistics, and projective geometry of Galois fields, § 5. Adiabatic analysis of remainders of geometric progressions, MCCMO, Moscow, 2006. (Russian)
Vladimir I. Arnold, Empirical study of stochasticity for deterministic chaotical dynamics of geometric progressions of residues, Funct. Anal. Other Math. 2 (2009), no. 2-4, 139–149. MR 2506112, DOI 10.1007/s11853-009-0034-7
—, How random are fractional parts of arithmetic progressions?, Uspekhi Mat. Nauk, 63 (2008), no. 2, 5–20; English transl. in Russian Math. Surveys 63 (2008), no. 2.
V. I. Arnol′d, Weak asymptotics of the numbers of solutions of Diophantine equations, Funktsional. Anal. i Prilozhen. 33 (1999), no. 4, 65–66 (Russian); English transl., Funct. Anal. Appl. 33 (1999), no. 4, 292–293 (2000). MR 1746430, DOI 10.1007/BF02467112
R. O. Kuzmin, On a problem of Gauss, Dokl. AN SSSR, Ser. A, 1928, 375–380. (See also Sur un Problème de Gauss. Atti Congr. Intern. Bologne. 1928. vol. 6. 83–99.)
A. Ja. Hinčin, Tsepnye drobi, 4th ed., “Nauka”, Moscow, 1978 (Russian). MR 514845
H. Gylden, Quelques remarques rélativement à la représentation des nombres irrationelles par des fractions continues, C. R. Acad. Sci. Paris, 107 (1888), 1584–1587.
M. O. Avdeeva, On the statistics of partial quotients of finite continued fractions, Funktsional. Anal. i Prilozhen. 38 (2004), no. 2, 1–11, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 38 (2004), no. 2, 79–87. MR 2086623, DOI 10.1023/B:FAIA.0000034038.55276.7d
V. Arnold, Arithmetics of binary quadratic forms, symmetry of their continued fractions and geometry of their de Sitter world, Bull. Braz. Math. Soc. (N.S.) 34 (2003), no. 1, 1–42. Dedicated to the 50th anniversary of IMPA. MR 1991436, DOI 10.1007/s00574-003-0001-8
V. I. Arnol′d, Statistics of the periods of continued fractions for quadratic irrationals, Izv. Ross. Akad. Nauk Ser. Mat. 72 (2008), no. 1, 3–38 (Russian, with Russian summary); English transl., Izv. Math. 72 (2008), no. 1, 1–34. MR 2394969, DOI 10.1070/IM2008v072n01ABEH002389
Francesca Aicardi, Empirical estimates of the average orders of orbits period lengths in Euler groups, C. R. Math. Acad. Sci. Paris 339 (2004), no. 1, 15–20 (English, with English and French summaries). MR 2075226, DOI 10.1016/j.crma.2004.02.021
Vladimir I. Arnold, Smooth functions statistics, Funct. Anal. Other Math. 1 (2006), no. 2, 111–118. MR 2385493, DOI 10.1007/s11853-007-0008-6
Liviu I. Nicolaescu, Morse functions statistics, Funct. Anal. Other Math. 1 (2006), no. 1, 85–91. MR 2381964, DOI 10.1007/s11853-007-0006-8
Vladimir I. Arnold, Statistics of the period lengths of the continued fractions for the eigenvalues of the integer matrices of order two, Funct. Anal. Other Math. 2 (2008), no. 1, 15–26. MR 2466084, DOI 10.1007/s11853-008-0013-4
Hiroyasu Tsuchihashi, Higher-dimensional analogues of periodic continued fractions and cusp singularities, Tohoku Math. J. (2) 35 (1983), no. 4, 607–639. MR 721966, DOI 10.2748/tmj/1178228955
E. I. Korkina, Two-dimensional continued fractions. The simplest examples, Trudy Mat. Inst. Steklov. 209 (1995), no. Osob. Gladkikh Otobrazh. s Dop. Strukt., 143–166 (Russian). MR 1422222
Elena Korkina, La périodicité des fractions continues multidimensionnelles, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 8, 777–780 (French, with English and French summaries). MR 1300940
O. N. Karpenkov, On examples of two-dimensional continued fractions, Cahiers du CEREMADE, Université Paris-Dauphine, 2004. No, 0430. 18 pp.
M. L. Kontsevich and Yu. M. Suhov, Statistics of Klein polyhedra and multidimensional continued fractions, Pseudoperiodic topology, Amer. Math. Soc. Transl. Ser. 2, vol. 197, Amer. Math. Soc., Providence, RI, 1999, pp. 9–27. MR 1733869, DOI 10.1090/trans2/197/02
V. Arnold, Statistics of Young diagrams of cycles of dynamical systems for finite tori automorphisms, Mosc. Math. J. 6 (2006), no. 1, 43–56, 221 (English, with English and Russian summaries). MR 2265946, DOI 10.17323/1609-4514-2006-6-1-43-56
—, Experimental observation of mathemaitcal objects, Summer School “Modern Mathematics”, Dubna, 2005. MCCMO, Moscow, 2006. (Russian)
V. L. Goncharov, On a topic of combinatorial analysis, Izvestiya AN SSSR, Ser. Matem. 8 (1944), 3–48. (Russian)
Ian Percival and Franco Vivaldi, Arithmetical properties of strongly chaotic motions, Phys. D 25 (1987), no. 1-3, 105–130. MR 887460, DOI 10.1016/0167-2789(87)90096-0
V. I. Arnol′d, Statistics of integral convex polygons, Funktsional. Anal. i Prilozhen. 14 (1980), no. 2, 1–3 (Russian). MR 575199
N. V. Smirnov, On estimates of divergence of two empirical distribution curves for two independent samples, Bull. Moskovsk. Universiteta, Matematika, 2 (1939), 3–14. (Russian)