Mathematics

In this paper we apply a homologous version of the Cauchy integral formula for octonionic monogenic functions to introduce for this class of functions the notion of multiplicity of zeroes and a -points in the sense of the topological mapping degree. As a big novelty we also address the case of zeroes lying on certain classes of compact zero varieties. This case has not even been studied in the associative Clifford analysis setting so far. We also prove an argument principle for octonionic monogenic functions for isolated zeroes and for non-isolated compact zero sets. In the isolated case we can use this tool to prove a generalized octonionic Rouché's theorem by a homotopic argument. As an application we set up a generalized version of Hurwitz theorem which is also a novelty even for the Clifford analysis case.

We establish a sharp upper bound for the absolute value of the derivative of the finite Blaschke product, provided that the critical values of this product lie in a given disk.

We consider Banach spaces E of functions holomorphic on the open unit disc D such that the unilateral shift S and the backward shift T are bounded on E. Assuming that the spectra of S and T are equal to the closed unit disc we discuss the existence of closed z-invariant of N of E having the "division property", which means that the function f λ : z → f (z)/ z-- λ belongs to N for every λ ∈ D and for every f ∈ N such that f ( λ ) = 0. This question is related to the existence of nontrivial bi-invariant subspaces of Banach spaces of hyperfunctions on the unit circle T.

Complex linear differential equations with entire coefficients are studied in the situation where one of the coefficients is an exponential polynomial and dominates the growth of all the other coefficients. If such an equation has an exponential polynomial solution f , then the order of f and of the dominant coefficient are equal, and the two functions possess a certain duality property. The results presented in this paper improve earlier results by some of the present authors, and the paper adjoins with two open problems.

A class of integral transforms, on the planar Gaussian Hilbert space with range in the weighted Bergman space on the bi-disk, is defined as the dual transforms of the 2d fractional Fourier transform associated with the Mehler function for Itô--Hermite polynomials. Some spectral properties of these transforms are investigated. Namely, we study their boundedness and identify their null spaces as well as their ranges. Such identification depends on the zeros set of Itô--Hermite polynomials. Moreover, the explicit expressions of their singular values are given and compactness and membership in p-Schatten class are studied. The relationship to specific fractional Hankel transforms is also established

We study the dynamics of a generic endomorphism f of an Oka-Stein manifold X . Such manifolds include all connected linear algebraic groups and, more generally, all Stein homogeneous spaces of complex Lie groups. We give several descriptions of the Fatou set and the Julia set of f . In particular, we show that the Julia set is the derived set of the set of attracting periodic points of f and that it is also the closure of the set of repelling periodic points of f . Among other results, we prove that f is chaotic on the Julia set and that every periodic point of f is hyperbolic. We also give an explicit description of the "Conley decomposition" of X induced by f into chain-recurrence classes and basins of attractors. For X=C , we prove that every Fatou component is a disc and that every point in the Fatou set is attracted to an attracting cycle or lies in a dynamically bounded wandering domain (whether such domains exist is an open question).

In this paper, we give a description of a natural invariant measure associated with a finitely generated polynomial semigroup (which we shall call the Dinh--Sibony measure) in terms of potential theory. This requires the theory of logarithmic potentials in the presence of an external field, which, in our case, is explicitly determined by the choice of a set of generators. Along the way, we establish the continuity of the logarithmic potential for the Dinh--Sibony measure, which might be of independent interest. We then use the F -functional of Mhaskar and Saff to discuss bounds on the capacity and diameter of the Julia sets of such semigroups.

We generalize our elliptic characterization of Oka manifolds to Oka maps. The generalized characterization can be considered as an affirmative answer to the relative version of Gromov's conjecture. As an application, we unify previously known Oka principles for submersions; namely the Gromov type Oka principle for subelliptic submersions and the Forstnerič type Oka principle for holomorphic fiber bundles with CAP fibers. We also establish the localization principle for Oka maps which gives new examples of Oka maps.

We study branching mappings that satisfy some condition of distortion of the modulus of families of paths. In a situation where the definition domain of mappings is locally connected on its boundary, the mapped domain is regular, and the majorant responsible for distortion of the modulus of families of paths is integrable, it is proved that the families of all specified mappings with one normalization condition are equicontinuous in the closure of the given domain.

We consider an embedded n -dimensional compact complex manifold in n+d dimensional complex manifolds. We are interested in the holomorphic classification of neighborhoods as part of Grauert's formal principle program. We will give conditions ensuring that a neighborhood of C n in M n+d is biholomorphic to a neighborhood of the zero section of its normal bundle. This extends Arnold's result about neighborhoods of a complex torus in a surface. We also prove the existence of a holomorphic foliation in M n+d having C n as a compact leaf, extending Ueda's theory to the high codimension case. Both problems appear as a kind linearization problem involving small divisors condition arising from solutions to their cohomological equations.