Motivated by a problem in complex dynamics, we examine the block structure of the natural action of iterated monodromy groups on the tree of preimages of a generic point. We show that in many cases, including when the polynomial has prime power degree, there are no large blocks other than those arising naturally from the tree structure. However, using a method of construction based on real grap...

The isomonodromic tau function defined by Jimbo-Miwa-Ueno vanishes on the Malgrange’s divisor of generalized monodromy data for which a vector bundle is nontrivial, or, which is the same, a certain Riemann–Hilbert problem has no solution. In their original work, Jimbo, Miwa, Ueno provided an algebraic construction of its derivatives with respect to isomonodromic times. However the dependence on...

Our aim is to find a general approach to the theory of classical solutions of the Garnier system in n-variables, Gn, based on the RiemannHilbert problem and on the geometry of the space of isomonodromy deformations. Our approach consists in determining the monodromy data of the corresponding Fuchsian system that guarantee to have a classical solution of the Garnier system Gn. This leads to the ...

Our aim is to find a general approach to the theory of classical solutions of the Garnier system in n-variables, Gn, based on the RiemannHilbert problem and on the geometry of the space of isomonodromy deformations. Our approach consists in determining the monodromy data of the corresponding Fuchsian system that guarantee to have a classical solution of the Garnier system Gn. This leads to the ...

In the present paper we consider some examples relative to the Deligne-Simpson problem (DSP) which is formulated like this: Give necessary and sufficient conditions upon the choice of the p+ 1 conjugacy classes cj ⊂ gl(n,C), resp. Cj ⊂ GL(n,C), so that there exist irreducible (p+1)-tuples of matrices Aj ∈ cj , A1 + . . .+Ap+1 = 0, resp. of matrices Mj ∈ Cj , M1 . . .Mp+1 = I. By definition, the...

For the fields depending on two of the four space-time coordinates only, the spaces of local solutions of various integrable reductions of Einstein’s field equations are shown to be the subspaces of the spaces of local solutions of the “null-curvature” equations constricted by a requirement of a universal (i.e. solution independent) structures of the canonical Jordan forms of the unknown matrix...

A new development of the “monodromy transform” method for analysis of hyperbolic as well as elliptic integrable reductions of Einstein equations is presented. Compatibility conditions for some alternative representations of the fundamental solutions of associated linear systems with spectral parameter in terms of a pair of dressing (“scattering”) matrices give rise to a new set of linear (quasi...