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Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)



Stochastic and deterministic characteristics of orbits in chaotically looking dynamical systems

Author: V. I. Arnold
Translated by: O. Khleborodova
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 70 (2009).
Journal: Trans. Moscow Math. Soc. 2009, 31-69
MSC (2000): Primary 37A45
Published electronically: December 3, 2009
MathSciNet review: 2573637
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Abstract | References | Similar Articles | Additional Information


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.

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Additional Information

V. I. Arnold
Affiliation: Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

Published electronically: December 3, 2009
Additional Notes: The work was partially supported by RFFI, Grant 05-01-00104.
Article copyright: © Copyright 2009 American Mathematical Society