B. Azevedo Scardua. Integration of Complex Differential Equations. J. Dynam. Control Systems 5 (1999), no. 1, 1--50.
Let X be a polynomial vector field in C^2; then it defines an algebraic foliation F on CP(2). If \fa admits a Liouvillian first integral on CP(2), then it is transversely affine outside some algebraic invariant curve S \subset CP(2). If, moreover, for some irreducible component S_0 \subset S, the singularities q \in Sing F \cap S are generic, then either F is given by a closed rational 1-form or it is a rational pull-back from a Bernoulli foliation R: p(x) dy - (y^2a(x)+yb(x)) dx=0 on \overline{C}\times \overline{C}. This result has several applications such as the study of foliations with algebraic limit sets on CP(2), the classification polynomial complete vector fields over C^2, and topological rigidity of foliations on CP(2). We also address the problem of moderate integration for germs of complex ordinary differential equations.
V.P. Kostov. Quantum States of Monodromy Group. J. Dynam. Control Systems 5 (1999), no. 1, 51--100.
Any irreducible finitely generated matrix group (with generators M_1, ... ,M_{p+1} satisfying the only relation M_1 ... M_{p+1}=I) is the monodromy group of some fuchsian linear system on Riemann's sphere. The eigenvalues of the matrices M_j define \lambda _{k,j}, the eigenvalues of the matrices-residues of the system only up to integers. There are always infinitely many possible choices of \lambda _{k,j}, a priori they must satisfy the only condition that their sum is 0. However, not always all a priori possible choices can be made. Some of them can be impossible due to the positions of the poles. Consider the a priori possible choices when the eigenvalues of only one matrix-residuum change (we presume that its pole is at 0). We show that infinitely many a priori possible choices are impossible if and only if the fuchsian system is obtained from another fuchsian system with a smaller number of poles and with a pole at 0 by the change of time t -> t^k/(p_kt^k+p_{k-1}t^{k-1}+ ... +p_0), p_i \in C, p_0 \neq 0, k>1. The result is applied to the Riemann--Hilbert problem.
A.A. Glutsuk. Stokes Operators Via Limit Monodromy of Generic Perturbations. J. Dynam. Control Systems 5 (1999), no. 1, 101--135.
We show that for a generic deformation of a linear analytic differential equation with an irregular singularity of order 1 of a generic (nonresonant) type, Stokes operators of the nonperturbed equation are limits of transition operators between appropriate eigenbases of the monodromy operators of the perturbed equation. We prove a generalization of this statement for arbitrary degree nonresonant irregular singularity.
A. Bufetov. Topological Entropy of Free Semigroup Actions and Skew-Products. J. Dynam. Control Systems 5 (1999), no. 1, 137--142.
A definition of topological entropy for a free semigroup action is suggested. Suppose that a free semigroup acts on a compact metric space by continuous self-maps. To this action, we assign a skew-product transformation whose fiber entropy is taken to be the entropy of the initial action. The main result is Theorem 1, a topological analogue of the Abramov--Rokhlin formula.
V.V. Ryzhikov. Transformations Having Homogeneous Spectra . J. Dynam. Control Systems 5 (1999), no. 1, 145--148.
We show that for a generic automorphism T, the Cartesian product T \times T has homogeneous spectrum of multiplicity two. New examples of automorphisms with the property \sigma \perp \sigma * \sigma are presented.
O.N. Ageev. On Ergodic Transformations with Homogeneous Spectrum . J. Dynam. Control Systems 5 (1999), no. 1, 149--152.
Rokhlin's problem on the existence of an ergodic transformation having a homogeneous spectrum of a finite multiplicity is solved. Katok's question about the spectral multiplicity function of Cartesian powers for a generic transformations is also answered.
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