The root extraction problem

The Nth root extraction problem for germs of diffeomorphisms is the problem of finding a germ of diffeomorphism such that gN=f, where gN is the Nth iterate of g under composition. Depending on f and on the multiplier of g at the origin there can be formal and analytic obstructions to a solution of the problem. By considering an unfolding of f we explain these obstructions. Indeed each analytic obstruction corresponds to an accumulation of periodic points which, in turn, are an obstruction to taking an Nth root of the unfolding. We apply this to the problem of the section of a curvilinear angle in N equal parts in conformal geometry.     

On the period function for a family of complex differential equations

We consider planar differential equations of the form being f(z) and g(z) holomorphic functions and prove that if g(z) is not constant then for any continuum of period orbits the period function has at most one isolated critical period, which is a minimum. Among other implications, the paper extends a well-known result for meromorphic equations, that says that any continuum of periodic orbits has a constant period function.     

Isochronous centers of Lienard type equations and applications

In this work we study Eq. (E) with a center at 0 and investigate conditions of its isochronicity. When f and g are analytic (not necessary odd) a necessary and sufficient condition for the isochronicity of 0 is given. This approach allows us to present an algorithm for obtained conditions for a point of (E) to be an isochronous center. In particular, we find again by another way the isochrones of the quadratic Loud systems (LD,F). We also classify a 5-parameters family of reversible cubic systems with isochronous centers.     

The period function of the generalized Lotka-Volterra centers

The present paper deals with the period function of the quadratic centers. In the literature different terminologies are used to classify these centers, but essentially there are four families: Hamiltonian, reversible , codimension four Q₄ and generalized Lotka-Volterra systems . Chicone [C. Chicone, Review in MathSciNet, Ref. 94h:58072] conjectured that the reversible centers have at most two critical periods, and that the centers of the three other families have a monotonic period function. With regard to the second part of this conjecture, only the monotonicity of the Hamiltonian and Q₄ families [W.A. Coppel, L. Gavrilov, The period function of a Hamiltonian quadratic system, Differential Integral Equations 6 (1993) 1357-1365; Y. Zhao, The monotonicity of period function for codimension four quadratic system Q₄, J. Differential Equations 185 (2002) 370-387] has been proved. Concerning the family, no substantial progress has been made since the middle 80s, when several authors showed independently the monotonicity of the classical Lotka-Volterra centers [F. Rothe, The periods of the Volterra-Lokta system, J. Reine Angew. Math. 355 (1985) 129-138; R. Schaaf, Global behaviour of solution branches for some Neumann problems depending on one or several parameters, J. Reine Angew. Math. 346 (1984) 1-31; J. Waldvogel, The period in the Lotka-Volterra system is monotonic, J. Math. Anal. Appl. 114 (1986) 178-184]. By means of the first period constant one can easily conclude that the period function of the centers in the family is monotone increasing near the inner boundary of its period annulus (i.e., the center itself). Thus, according to Chicone's conjecture, it should be also monotone increasing near the outer boundary, which in the Poincaré disc is a polycycle. In this paper we show that this is true. In addition we prove that, except for a zero measure subset of the parameter plane, there is no bifurcation of critical periods from the outer boundary. Finally we show that the period function is globally (i.e., in the whole period annulus) monotone increasing in two other cases different from the classical one.     

Proper analytic free maps

This paper concerns analytic free maps. These maps are free analogs of classical analytic functions in several complex variables, and are defined in terms of non-commuting variables amongst which there are no relations - they are free variables. Analytic free maps include vector-valued polynomials in free (non-commuting) variables and form a canonical class of mappings from one non-commutative domain D in say g variables to another non-commutative domain in variables. As a natural extension of the usual notion, an analytic free map is proper if it maps the boundary of D into the boundary of . Assuming that both domains contain 0, we show that if is a proper analytic free map, and f(0)=0, then f is one-to-one. Moreover, if also , then f is invertible and f−1 is also an analytic free map. These conclusions on the map f are the strongest possible without additional assumptions on the domains D and .     

Complex hyperbolic Fenchel-Nielsen coordinates

Let Σ be a closed, orientable surface of genus g. It is known that the representation variety of π₁(Σ) has 2g−3 components of (real) dimension 16g−16 and two components of dimension 8g−6. Of special interest are the totally loxodromic, faithful (that is quasi-Fuchsian) representations. In this paper we give global real analytic coordinates on a subset of the representation variety that contains the quasi-Fuchsian representations. These coordinates are a natural generalisation of Fenchel-Nielsen coordinates on the Teichmüller space of Σ and complex Fenchel-Nielsen coordinates on the (classical) quasi-Fuchsian space of Σ.     

Linear pencils on real algebraic curves

Our knowledge of linear series on real algebraic curves is still very incomplete. In this paper we restrict to pencils (complete linear series of dimension one). Let X denote a real curve of genus g with real points and let k(R) be the smallest degree of a pencil on X (the real gonality of X). Then we can find on X a base point free pencil of degree g+1 (resp. g if X is not hyperelliptic, i.e. if k(R)>2) with an assigned geometric behaviour w.r.t. the real components of X, and if we prove that which is the same bound as for the gonality of a complex curve of even genus g. Furthermore, if the complexification of X is a k-gonal curve (k≥2) one knows that k≤k(R)≤2k−2, and we show that for any two integers k≥2 and 0≤n≤k−2 there is a real curve with real points and k-gonal complexification such that its real gonality is k+n.     

Extending homeomorphisms from punctured surfaces to handlebodies

Let Hg be a genus g handlebody and MCG2n(Tg) be the group of the isotopy classes of orientation preserving homeomorphisms of Tg=∂Hg, fixing a given set of 2n points. In this paper we find a finite set of generators for , the subgroup of MCG2n(Tg) consisting of the isotopy classes of homeomorphisms of Tg admitting an extension to the handlebody and keeping fixed the union of n disjoint properly embedded trivial arcs. This result generalizes a previous one obtained by the authors for n=1. The subgroup turns out to be important for the study of knots and links in closed 3-manifolds via (g,n)-decompositions. In fact, the links represented by the isotopy classes belonging to the same left cosets of in MCG2n(Tg) are equivalent.     

Simultaneous bifurcation of limit cycles from two nests of periodic orbits

Let be an holomorphic differential equation having a center at p, and consider the following perturbation . We give an integral expression, similar to an Abelian integral, whose zeroes control the limit cycles that bifurcate from the periodic orbits of the period annulus of p. This expression is given in terms of the linearizing map of at p. The result is applied to control the simultaneous bifurcation of limit cycles from the two period annuli of , after a polynomial perturbation.     

New Pólya-Schoenberg type theorems

In this paper the theory of Hadamard product multipliers is extended from the unit disk in the complex plane to arbitrary so-called disk-like domains, i.e. such domains which are the union of disks or half-planes, all containing the origin. In such a domain, say Ω, we define (the class of) generalized prestarlike functions of order α?1 and ask for Hadamard multipliers g analytic at z=0 for which implies . We prove that such a multiplier necessarily has to be analytic in

     

Refinements of Milnor's fibration theorem for complex singularities

Let X be an analytic subset of an open neighbourhood U of the origin in Cn. Let be holomorphic and set V=f−1(0). Let Bε be a ball in U of sufficiently small radius ε>0, centred at . We show that f has an associated canonical pencil of real analytic hypersurfaces Xθ, with axis V, which leads to a fibration Φ of the whole space (X∩Bε)?V over S¹. Its restriction to (X∩Sε)?V is the usual Milnor fibration , while its restriction to the Milnor tube f−1(∂Dη)∩Bε is the Milnor-Lê fibration of f. Each element of the pencil Xθ meets transversally the boundary sphere Sε=∂Bε, and the intersection is the union of the link of f and two homeomorphic fibres of ? over antipodal points in the circle. Furthermore, the space obtained by the real blow up of the ideal (Re(f),Im(f)) is a fibre bundle over RP¹ with the Xθ as fibres. These constructions work also, to some extent, for real analytic map-germs, and give us a clear picture of the differences, concerning Milnor fibrations, between real and complex analytic singularities.     

Reflection about k-jets and holomorphic extension of CR mappings

Let be a CR mapping between real analytic generic submanifolds M, M₁ of and , respectively. According to Webster's theory (Proc. Amer. Math. Soc. 86 (1982) 236-240) and its further developments, f has holomorphic extension to a full neighborhood of M in when the following requirements are fulfilled: f extends to a wedge W continuous up to M; f is of class C^k; (where denotes the complex tangent bundle); M₁ is “k-nondegenerate.” We deal here with the case where is strictly smaller than but is still real analytic in suitable sense. We show that a suitably refined condition of k-nondegeneracy still entails holomorphic extension of f.     

Vertex operator algebras associated to modified regular representations of affine Lie algebras

Let G be a simply-connected complex Lie group with simple Lie algebra g and let be its affine Lie algebra. We use intertwining operators and Knizhnik-Zamolodchikov equations to construct a family of N-graded vertex operator algebras (VOAs) associated to g. These vertex operator algebras contain the algebra of regular functions on G as the conformal weight 0 subspaces and are -modules of dual levels in the sense that , where h^∨ is the dual Coxeter number of g. This family of VOAs was previously studied by Arkhipov-Gaitsgory and Gorbounov-Malikov-Schechtman from different points of view. We show that when k is irrational, the vertex envelope of the vertex algebroid associated to G and the level k is isomorphic to the vertex operator algebra we constructed above. The case of rational levels is also discussed.     

Approximation by smooth functions with no critical points on separable Banach spaces

We characterize the class of separable Banach spaces X such that for every continuous function and for every continuous function there exists a C¹ smooth function for which |f(x)−g(x)|?ε(x) and g^′(x)≠0 for all x∈X (that is, g has no critical points), as those infinite-dimensional Banach spaces X with separable dual X^∗. We also state sufficient conditions on a separable Banach space so that the function g can be taken to be of class Cp, for p=1,2,…,+∞. In particular, we obtain the optimal order of smoothness of the approximating functions with no critical points on the classical spaces ?p(N) and Lp(Rn). Some important consequences of the above results are (1) the existence of a non-linear Hahn-Banach theorem and the smooth approximation of closed sets, on the classes of spaces considered above; and (2) versions of all these results for a wide class of infinite-dimensional Banach manifolds.     

Bifurcations and distribution of limit cycles which appear from two nests of periodic orbits

In this paper, we study the distribution and simultaneous bifurcation of limit cycles bifurcated from the two periodic annuli of the holomorphic differential equation , after a small polynomial perturbation. We first show that, under small perturbations of the form , where is a polynomial of degree 2m−1 in which the power of z is odd and the power of is even, the only possible distribution of limit cycles is (u,u) for all values of u=0,1,2,…,m−3. Hence, the sharp upper bound for the number of limit cycles bifurcated from each two period annuli of is m−3, for m≥4. Then we consider a perturbation of the form , where is a polynomial of degree m in which the power of z is odd and obtain the upper bound m−5, for m≥6. Moreover, we show that the distribution (u,v) of limit cycles is possible for 0≤u≤m−5, 0≤v≤m−5 with u+v≤m−2 and m≥9.