Stochastic and Deterministic Characteristics of Orbits in Chaotically Looking Dynamical Systems

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. 1. Kolmogorov’s stochasticity parameter Laplace was calculating the probability that the sun will rise tomorrow. It is not easy to determine what he meant by “probability”. First attempts to define an objectively measurable degree of randomness of observable events were made by von Mises and Kolmogorov. In his 1933 Italian paper on actuarial mathematics (see Kolmogoroff [1]), Kolmogorov introduced his “stochasticity parameter” λn of a set of n real numbers, which allows one to estimate how realistic is the assumption that these n numbers are n independent values of the same real random variable. The definition of this stochasticity parameter λn begins with the description of the given set of n real numbers using the “empirical counting function” Cn defined below. Let us arrange the given numbers in increasing order, x1 ≤ x2 ≤ · · · ≤ xn. The value of the empirical counting function Cn(X) is defined as the number of elements xm that are not larger than X: Cn(X) = 0 for X < x1; Cn(X) = m for xm ≤ X < xm+1; Cn(X) = n for xn ≤ X. Kolmogorov assumed that the random variable x ∈ R has a continuous distribution function and describes the “theoretical counting function” C0(X) as the expectation of 2000 Mathematics Subject Classification. Primary 37A45. The work was partially supported by RFFI, Grant 05-01-00104. c ©2009 American Mathematical Society