Inverse problems of symbolic dynamics

This paper reviews some results regarding symbolic dynamics, correspondence between languages of dynamical systems and combinatorics. Sturmian sequences provide a pattern for investigation of one-dimensional systems, in particular interval exchange transformation. Rauzy graphs language can express many important combinatorial and some dynamical properties. In this case combinatorial properties are considered as being generated by substitutional system, and dynamical properties are considered as criteria of superword being generated by interval exchange transformation. As a consequence, one can get a morphic word appearing in interval exchange transformation such that frequencies of letters are algebraic numbers of an arbitrary degree. Concerning multydimensional systems, our main result is the following. Let P(n) be a polynomial, having an irrational coefficient of the highest degree. A word $w$ (w=(w_n), n\in \nit) consists of a sequence of first binary numbers of $\{P(n)\}$ i.e. $w_n=[2\{P(n)\}]$. Denote the number of different subwords of $w$ of length $k$ by $T(k)$ . \medskip {\bf Theorem.} {\it There exists a polynomial $Q(k)$, depending only on the power of the polynomial $P$, such that $T(k)=Q(k)$ for sufficiently great $k$.

Subexponential estimations in Shirshov's height theorem (in English)

In 1993 E. I. Zelmanov asked the following question in Dniester Notebook: "Suppose that F_{2, m} is a 2-generated associative ring with the identity x^m=0. Is it true, that the nilpotency degree of F_{2, m} has exponential growth?" We show that the nilpotency degree of l-generated associative algebra with the identity x^d=0 is smaller than Psi(d,d,l), where Psi(n,d,l)=2^{18} l (nd)^{3 log_3 (nd)+13}d^2. We give the definitive answer to E. I. Zelmanov by this result. It is the consequence of one fact, which is based on combinatorics of words. Let l, n and d>n be positive integers. Then all the words over alphabet of cardinality l which length is greater than Psi(n,d,l) are either n-divided or contain d-th power of subword, where a word W is n-divided, if it can be represented in the following form W=W_0 W_1...W_n such that W_1 >' W_2>'...>'W_n. The symbol >' means lexicographical order here. A. I. Shirshov proved that the set of non n-divided words over alphabet of cardinality l has bounded height h over the set Y consisting of all the words of degree <n. Original Shirshov's estimation was just recursive, in 1982 double exponent was obtained by A.G.Kolotov and in 1993 A.Ya.Belov obtained exponential estimation. We show, that h<Phi(n,l), where Phi(n,l) = 2^{87} n^{12 log_3 n + 48} l. Our proof uses Latyshev idea of Dilworth theorem application.Comment: 21 pages, Russian version of the article is located at the link arXiv:1101.4909; Sbornik: Mathematics, 203:4 (2012), 534 -- 55

Long-period cosmic ray variations and their altitude dependence

Long-period variations were studied from the data of ground-based cosmic ray (CR) observations. In spite of a large value of an 2-year variation, it is more difficult to obtain its spectrum than the spectrum of a solar diurnal variation. Serious obstacles are caused by changes in individual detectors and in the whole world wide network of CR detectors, by the absence of continuity and uniformity of data series, by various apparatus variations. In discrimination and investigation of long-period variations an important and determining point is preparation and preliminary analysis of data

Solar activity beyond the disk and variations of the cosmic ray gradient

Part of galactic cosmic rays (CR) observed near the Earth and on the Earth come from beyond-disk regions of circumsolar space. But CR of those energies which undergo substantial modulation cover too large a path across the lines of force of the interplanetary magnetic field (IMF) in order that they could provide an effective transfer of information about beyond-disk solar activity. And if it is still possible, the most probable channel for transferring such information must be a neutral layer of heliomagnetosphere in which the transverse CR transport is facilitated by their drift in an inhomogeneous magnetic field. A simple diffusion model for an expected CR variation in a neutral layer near the Earth is discussed. It is of importance that variations of the CR gradient are not at all always accompanied by considerable variations of IMF and solar wind velocity at the point of observation