This chapter is a survey of several relevant systems to which the grouptheoretic scheme of the preceding chapters or its modifications can be applied. The choice of topics for this chapter was intended to show different (but nevertheless, “hydrodynamic”) features of a variety of dynamical systems and to emphasize suggestive points for further study and future results.

The group we will most often be dealing with in hydrodynamics is the infinite-dimensional group of diffeomorphisms that preserve the volume element of the domain of a fluid flow.

Stars and planets possess magnetic fields that permanently change. Earth, for instance, mysteriously interchanges its north and south magnetic poles, so that the time pattern of the switches forms a Cantor-type set on the time scale (see [AnS]). The mechanism of generation of magnetic fields in astrophysical objects (or in electrically conducting fluids) constitutes the subject of dynamo theory. Kinematic dynamo theory studies what kind of fluid motion can induce exponential growth of a magnetic field for small magnetic diffusivity. Avoiding analytical and numerical results (though crucial for this field), we address below the topological side of the theory.

In 1963 E. N. Lorenz stated that a two-week forecast would be the theoretical bound for predicting the future state of the atmosphere using large-scale numerical models [Lor]. Modern meteorology has currently reached a good correlation of the observed versus predicted for roughly seven days in the northern hemisphere, whereas this period is shortened by half in the southern hemisphere and by two-thirds in the tropics for the same level of correlation [Kri]. These differences are due to a very simple factor: the available data density.

The interior media of stars and planets are often virtually perfect conductors and possess magnetic fields. These fields are said to be “frozen" into the medium (for instance, plasma or magma) in spite of temperatures of a million degrees. Mathematically this means that any motion of the medium transports the fields by a diffeomorphism action preserving the mutual alocation of the fields' trajectories. Such a transform may diminish the field magnetic energy. The topological structure of the field provides obstacles to the full dissipation of the magnetic energy of the star or planet.

The complex Euler group is defined associating to an integer complex number z the multiplicative group of the complex integers residues modulo z , relatively prime to z . This group is calculated for z =(3+0 i ) n : it is isomorphic to the product of three cyclic group or orders (8, 3 n −1 and 3 n −1 ).

The Fermat–Euler progression of residues modulo n is a geometrical progression, formed by the powers of one residue, a. Such a sequence is always periodic, starting from some place. The object of study of the paper is the minimal period’s length T(a,n) of this progression.

The Gibbs phenomenon is described for the Fourier series of a function at its jump, the function being defined along the finite circle ℤ/pℤ.

Kolmogorov discovered in 1933 that the empirical statistics of several independent values of any random variable differs from the true distribution function of this variable in some universal way: the random distribution of the distance of one of these statistics from the other verifies (asymptotically) some stochastic distribution law (called later “Kolmogorov’s distribution”).

Empirical study of the period’s length T of the continued fractions of \(\sqrt{Q}\) (for growing integers Q) shows several strange asymptotical results, for instance, \(T\leq C\sqrt{Q}\ln{Q}\). These results show important differences between the statistics of the elements of the continued fractions of random real numbers and of square roots of random integers.

An algebraic permutation \(\hat{A}\in S(N=n^{m})\) is the permutation of the N points of the finite torus ℤ n m , realized by a linear operator A∈SL(m,ℤ n ). The statistical properties of algebraic permutations are quite different from those of random permutations of N points. For instance, the period length T(A) grows superexponentially with N for some (random) permutations A of N elements, whereas \(T(\hat{A})\) is bounded by a power of N for algebraic permutations \(\hat{A}\). The paper also contains a strange mean asymptotics formula for the number of points of the finite projective line P1(ℤ n ) in terms of the zeta function.

We study the asymptotic behavior of the period lengths of the continued fractions for the eigenvalues of integer matrices of order two belonging to 4-balls as the ball radius increases to infinity.

The article describes the interrelations between the minimal integer number N(a,b,c) which belongs to the additive semigroup of integers generated by a, b, c together with all greater integers, on the one hand, and the geometrical theory of continued fractions describing the convex hulls of sets of integer points in simplicial cones, on the other hand. It also provides some hints on the extension of N to non-integral arguments.

We present geometrical arguments suggesting that the part of the segment {0,1,…,N−1} covered by the additive semigroup generated by (a,b,c) between 0 and the Frobenius number N(a,b,c) should exceed λ V for some constant λ (which might be 1/3 or even more).

In his 16th problem, Hilbert asked to study the topological structure of the level lines of real polynomials of n variables. Our goal is to show that the topological classification of the real polynomials defining these real algebraic curves is a richer problem. For instance, there are 17746 smooth Morse functions on S 2 having T =4 saddles (the maximum value for the fourth-degree polynomials on ℝ 2 ). The ergodic theory of random graphs, basic for this study, suggests the growth rate T 2 T for a large number T of saddles, and we prove the lower and upper bounds of the orders T T and T 2 T for the number of topological types.