Algebras generated by two bounded holomorphic functions

We study the closure in the Hardy space or the disk algebra of algebras generated by two bounded functions, of which one is a finite Blaschke product. We give necessary and sufficient conditions for density or finite codimension of such algebras. The conditions are expressed in terms of the inner part of a function which is explicitly derived from each pair of generators. Our results are based on identifying z-invariant subspaces included in the closure of the algebra. Versions of these results for the case of the disk algebra are given.Comment: 22 pages ; a number of minor mistakes have been corrected, and some points clarified. Conditionally accepted by Journal d'Analyse Mathematiqu

Hyperholomorpic connections on coherent sheaves and stability

Let $M$ be a hyperkaehler manifold, and $F$ a torsion-free and reflexive coherent sheaf on $M$. Assume that $F$ (outside of its singularities) admits a connection with a curvature which is invariant under the standard SU(2)-action on 2-forms. If the curvature is square-integrable, then $F$ is stable and its singularities are hyperkaehler subvarieties in $M$. Such sheaves (called hyperholomorphic sheaves) are well understood. In the present paper, we study sheaves admitting a connection with SU(2)-invariant curvature which is not necessarily square-integrable. This situation arises often, for instance, when one deals with higher direct images of holomorphic bundles. We show that such sheaves are stable.Comment: 37 pages, version 11, reference updated, corrected many minor errors and typos found by the refere

Post-critical set and non existence of preserved meromorphic two-forms

We present a family of birational transformations in $CP_2$ depending on two, or three, parameters which does not, generically, preserve meromorphic two-forms. With the introduction of the orbit of the critical set (vanishing condition of the Jacobian), also called ``post-critical set'', we get some new structures, some "non-analytic" two-form which reduce to meromorphic two-forms for particular subvarieties in the parameter space. On these subvarieties, the iterates of the critical set have a polynomial growth in the \emph{degrees of the parameters}, while one has an exponential growth out of these subspaces. The analysis of our birational transformation in $CP_2$ is first carried out using Diller-Favre criterion in order to find the complexity reduction of the mapping. The integrable cases are found. The identification between the complexity growth and the topological entropy is, one more time, verified. We perform plots of the post-critical set, as well as calculations of Lyapunov exponents for many orbits, confirming that generically no meromorphic two-form can be preserved for this mapping. These birational transformations in $CP_2$, which, generically, do not preserve any meromorphic two-form, are extremely similar to other birational transformations we previously studied, which do preserve meromorphic two-forms. We note that these two sets of birational transformations exhibit totally similar results as far as topological complexity is concerned, but drastically different results as far as a more ``probabilistic'' approach of dynamical systems is concerned (Lyapunov exponents). With these examples we see that the existence of a preserved meromorphic two-form explains most of the (numerical) discrepancy between the topological and probabilistic approach of dynamical systems.Comment: 34 pages, 7 figure

Invariant Distances

In this chapter we shall define the (invariant) distance we are going to use, and collect some of its main properties we shall need later on

Green Currents for Meromorphic Maps of Compact K\"ahler Manifolds

We consider the dynamics of meromorphic maps of compact K\"ahler manifolds. In this work, our goal is to locate the non-nef locus of invariant classes and provide necessary and sufficient conditions for existence of Green currents in codimension one.Comment: Statement of Theorem 1.5 is slightly improved. Proposition 5.2 and Theorem 5.3 are adde

Birth after TESE–ICSI in a man with hypogonadotropic hypogonadism and congenital adrenal hypoplasia linked to a DAX-1 (NR0B1) mutation

DAX1/NR0B1 mutations are responsible for X-linked congenital adrenal hypoplasia (AHC) associated with hypogonadotropic hypogonadism (HH). Few data are available concerning testicular function and fertility in men with DAX1 mutations. Azoospermia as well as failure of gonadotrophin treatment have been reported. We induced spermatogenesis in a patient who has a DAX1 mutation (c.1210C>T), leading to a stop codon in position 404 (p.Gln404X). His endocrine testing revealed a low testosterone level at 1.2 nmol/l (N: 12–40) with low FSH and LH levels at 2.1 IU/l (N: 1–5 IU/l) and 0.1 IU/l (N: 1–4 IU/l), respectively. Baseline semen analysis revealed azoospermia. Menotropin (Menopur®:150 IU, three times weekly) and human chorionic gonadotrophin (1500 IU, twice weekly) were used. After 20 months of treatment, as azoospermia persisted, bilateral multiple site testicular biopsies were performed. Histology revealed severe hypospermatogenesis. Rare spermatozoa were extracted from the right posterior fragment and ICSI was performed. Four embryos were obtained and, after a frozen–thawed single-embryo transfer, the patient's wife became pregnant and gave birth to a healthy boy. We report the first case of paternity after TESE–ICSI in a patient with DAX1 mutation, giving potential hope to these patients to father non-affected children. Furthermore, this case illustrates the fact that patients with X-linked AHC have a primary testicular defect in addition to HH

Lyapunov exponents, bifurcation currents and laminations in bifurcation loci

Bifurcation loci in the moduli space of degree $d$ rational maps are shaped by the hypersurfaces defined by the existence of a cycle of period $n$ and multiplier 0 or $e^{i\theta}$. Using potential-theoretic arguments, we establish two equidistribution properties for these hypersurfaces with respect to the bifurcation current. To this purpose we first establish approximation formulas for the Lyapunov function. In degree $d=2$, this allows us to build holomorphic motions and show that the bifurcation locus has a lamination structure in the regions where an attracting basin of fixed period exists