A characterization of Fuchsian groups acting on complex hyperbolic spaces

Let G ? SU(2, 1) be a non-elementary complex hyperbolic Kleinian group. If G preserves a complex line, then G is ?-Fuchsian; if G preserves a Lagrangian plane, then G is ?-Fuchsian; G is Fuchsian if G is either ?-Fuchsian or ?-Fuchsian. In this paper, we prove that if the traces of all elements in G are real, then G is Fuchsian. This is an analogous result of Theorem V.G. 18 of B. Maskit, Kleinian Groups, Springer-Verlag, Berlin, 1988, in the setting of complex hyperbolic isometric groups. As an application of our main result, we show that G is conjugate to a subgroup of S(U(1)×U(1, 1)) or SO(2, 1) if each loxodromic element in G is hyperbolic. Moreover, we show that the converse of our main result does not hold by giving a ?-Fuchsian group.     

General linear problem of the isomonodromic deformation of Fuchsian systems

In contrast to nonresonance systems whose continuous deformations are always Schlesinger deformations, systems with resonances provide great possibilities for deformations. In this case, the number of continuous parameters of deformation, in addition to the location of the poles of the system, includes the data describing the Levelt structure of the system, or, in other words, the distribution of resonance directions in the space of solutions. The question of classifying the form and structure of deformations according to these parameters arises. In the present paper, we consider continuous isomonodromic deformations of Fuchsian systems, including those with respect to additional parameters, describe the corresponding linear problem, and present the Pfaff form of the linear problem of general continuous isomonodromic deformation of Fuchsian systems.     

A linear Fuchsian equation with variable indices

We construct three types of solutions for a Fuchsian equation with variable indices: (1) branched solutions involving logarithms of the time variable t; (2) solutions involving t^x, where x is a space variable; and, (3) for a model case, exact solutions involving hypergeometric functions. These three solutions have completely different singularities. The constructions are given in a form suitable for application to more general equations. As an illustration, we resolve in particular an apparent discrepancy between two recent results on this problem.     

Fiber spaces over Teichmüller spaces for Fuchsian groups with elliptic elements

We determine the biholomorphic fiber preserving isomorphisms of fiber spaces over Teichmüller spaces for Fuchsian groups with elliptic elements. We show that except in some special cases a biholomorphic fiber preserving isomorphism between two Bers fiber spaces is always an allowable mapping. We find that the situation is different for Teichmüller curves, showing that in general there are some other biholomorphic fiber preserving isomorphisms between Teichmüller curves besides the allowable mappings. Research supported by the National Natural Science Foundation of China.     

Fuchsian systems with completely reducible monodromy

The solvability of the Riemann-Hilbert problem for representations χ = χ ₁ ⊕ χ ₂ having the form of a direct sum is considered. It is proved that any representation χ ₁ can be realized as a direct summand in the monodromy representation χ of a Fuchsian system. Other results are also obtained, which suggest a simple method for constructing counterexamples to the Riemann-Hilbert problem.     

On constraint controllability of linear systems in Banach spaces

In this paper, we give an addendum to a result of Dolecki and Russell (Ref. 1) related to the duality relationship between observation and control for linear systems in Banach spaces. Our results relate the controllability of a system to the constraint controllability of that system and to the observability of an adjoint system. The main tool used here is an extension of the classical open mapping theorem.     

Robust stability of positive linear systems in Banach spaces

In this paper, we study the stability radii of positive linear systems with delays with respect to various classes of perturbations in infinite dimensional spaces. It is shown that the positive, real and complex stability radii coincide. Moreover, explicit formulas are derived for these stability radii and illustrated by a simple example.     

Two-dimensional Fuchsian systems and the Chebyshev property

Let (x(t),y(t))^? be a solution of a Fuchsian system of order two with three singular points. The vector space of functions of the form P(t)x(t)+Q(t)y(t), where P,Q are real polynomials, has a natural filtration of vector spaces, according to the asymptotic behavior of the functions at infinity. We describe a two-parameter class of Fuchsian systems, for which the corresponding vector spaces obey the Chebyshev property (the maximal number of isolated zeros of each function is less than the dimension of the vector space). Up to now, only a few particular systems were known to possess such a non-oscillation property. It is remarkable that most of these systems are of the type studied in the present paper. We apply our results in estimating the number of limit cycles that appear after small polynomial perturbations of several quadratic or cubic Hamiltonian systems in the plane.     

Middle convolution and deformation for Fuchsian systems

Middle convolution and addition are operations for Fuchsiansystems of differential equations which preserve the numberof accessory parameters. In this paper we show that they alsopreserve the deformation equations. Several Bäcklund transformationsof the sixth Painlevé equation are obtained from thisviewpoint.     

Dichotomy properties of Hamiltonians for linear systems in extrapolation spaces

For the Hamiltonian operator matrix from systems theory the existence of invariant subspaces corresponding to the spectrum in the right and left half-plane is shown. The control and observation operators are unbounded in the sense that they map into extrapolation spaces, thereby allowing for PDE systems with control and observation on the boundary. The invariant subspaces are then used to construct solutions of the corresponding Riccati equation. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Galois group of some Fuchsian systems

In this paper, we give a new result ofn the differential Galois theory of linear ordinary differential equations. In particular, we compute the differential Galois group for a special type of nonresonant Fuchsian system.     

Differential systems with Fuchsian linear part: Correction and linearization, normal forms and multiple orthogonal polynomials

Differential systems with a Fuchsian linear part are studied in regions including all the singularities in the complex plane of these equations. Such systems are not necessarily analytically equivalent to their linear part (they are not linearizable) and obstructions are found as a unique nonlinear correction after which the system becomes formally linearizable.More generally, normal forms are found.The corrections and the normal forms are found constructively. Expansions in multiple orthogonal polynomials and their generalization to matrix-valued polynomials are instrumental to these constructions.