135 research outputs found
Moduli of spherical tori with one conical point
No full textWe determine the topology of the moduli space MS1;1(v) of surfaces of genus one with a Riemannian metric of constant curvature 1 and one conical point of angle 2 pi v. In particular, for v is an element of (2m-1, 2m + 1) nonodd, MS1,1(v) is connected, has orbifold Euler characteristic -1/12m(2), and its topology depends on the integer m > 0 only. For v= 2m + 1 odd, MS1,1(v) has 1/6 m(m + 1) connected components. For v= 2m even, MS1,1(v) has a natural complex structure and it is biholomorphic to H-2/G(m) for a certain subgroup Gm of SL(2, Z) of index m(2), which is nonnormal for m > 1
An upper estimate for characteristic exponent of polynomials
No full textIn (10), A. Eremenko and G. Levin have found an upper bound for the characteristic exponent of polynomials with connected Julia set. In (11), they extended their result so that it includes the polynomials of the form P\sb{c}(z)=z\sp{d}+c. In the case of polynomials with connected Julia set, the upper bound is sharp, and in the second case it is asymptotically the best possible upper bound. In this paper we extend their result to all polynomials
Rational maps with real multipliers
No full textLet f be a rational function such that the multipliers of all repelling periodic points are real. We prove that the Julia set of such a function belongs to a circle. Combining this with a result of Fatou we conclude that whenever J(f) belongs to a smooth curve, it also belongs to a circle. Then we discuss rational functions whose Julia sets belong to a circle
Determining biholomorphic type of a manifold using combinatorial and algebraic structures
No full textWe settle two problems of reconstructing a biholomorphic type of a manifold. In the first problem we use graphs associated to Riemann surfaces of a particular class. In the second one we use the semigroup structure of analytic endomorphisms of domains in [special characters omitted]. 1. We give a new proof of a theorem due to P. Doyle. The problem is to determine a conformal type of a Riemann surface of class Fq, using properties of the associated Speiser graph. Sufficient criteria of type have been given since 1930\u27s when the class Fq was introduced. Also there were necassary and sufficient results which have theoretical value, but which are hard to apply. P. Doyle\u27s theorem states that a non-compact Riemann surface of class Fq has a hyperbolic (parabolic) type, if and only if its extended Speiser graph is hyperbolic (parabolic). By a hyperbolic graph we mean a locally-finite infinite connected graph, which admits a non-constant positive superharmonic function with respect to the discrete Laplace operator. Otherwise a graph is parabolic. The usefulness of this criterion stems from the possibility of applying Rayleigh\u27s short-cut method for graphs. We apply Doyle\u27s theorem to give a counterexample to a conjecture of R. Nevanlinna that relates the type to an excess of a Speiser graph. More explicitely, the conjecture was that if the (upper) mean excess of a surface of class Fq is negative, then the surface is hyperbolic. We provide an example of a parabolic surface of class Fq with negative mean excess. 2. If there is a biholomorphic or antibiholomorphic map between two domains in [special characters omitted], then it gives rise to an isomorphism between the semigroups of analytic endomorphisms of these domains. Suppose, conversely, that we are given two domains in [special characters omitted] with isomorphic semigroups of analytic endomorphisms. Are they biholomorphically or antibiholomorphically equivalent? This question was raised by L. Rubel. Similar questions were studied in the setting of topological spaces. The case n = 1 was investigated by A. Eremenko, who showed that if we require that the domains are bounded, then the answer to the above question is positive. It was shown by A. Hinkkanen that the boundedness condition cannot be dropped. We prove that two bounded domains in [special characters omitted] with isomorphic semigroups of analytic endomorphisms are biholomorphically or antibiholomorphically equivalent. Moreover, we generalize this by requiring only the existence of an epimorphism between the semigroups
Über Funktionen in der Speiser-Klasse mit einem Trakt
No full textLet f be a transcendental entire map. A complex number w is called critical value of f if there exists a complex number z such that f'(z)=0 and f(z)=w. A complex number b is called an asymptotic value of f if there exists a curve \gamma with \gamma(t)\to\infty as t\to\infty but f(\gamma(t))\to b as t\to\infty. The singular set of f is the set consisting of all critical and asymptotic values of f. The set B of all transcendental entire functions with a bounded singular set is called Eremenko-Lyubich class. The Speiser class S consists of all functions in class B where the singular set is not only bounded but finite.
These classes have been thoroughly studied, notably, in complex dynamics, which deals with the behaviour of an entire or rational map f under iteration. Of particular interest is the construction of functions in classes B and S with prescribed behaviour. One method to obtain maps in class B is using so-called Cauchy integrals. Gwyneth Stallard used this method to prove that for any d\in(1,2) there exists a function in class B whose Julia set has Hausdorff dimension equal to d. It is the shape of the tracts of her maps which yields the desired Hausdorff dimension. Here, a tract is a connected component of the set where the modulus of the function is large.
The question arises whether Stallard's result also holds for maps in class S. While we are not able to answer this question, we show that there exist functions in class S whose tracts are in some sense similar to the tracts used by Stallard. The method of Cauchy integrals does generally not generate maps in class S. An alternative construction method is quasiconformal folding, which was recently introduced by Christopher Bishop.
We use Bishop's method to construct quasiregular maps which only grow in one parabola shaped tract which is symmetric to the real axis and are bounded otherwise. Furthermore, we prove that for each constructed quasiregular map g there exists an entire function f in class S such that g=f\circ\phi for some quasiconformal homeomorphism \phi. Thus, the tract of f, which is still symmetric to the real axis, is the quasiconformal image of the tract of g. Moreover, the quasiconformal map involved is asymptotically conformal at infinity. We use this to prove that the maximum modulus M(r,f) of f on the circle with radius r is bounded below by a function which depends on the shape of the tract. In particular, we prove that there exists an entire map f in the class S with only one tract, which is symmetric to the real axis, such that \log\log M(r,f) is bounded below by d\cdot\sqrt{r} for some d>0.Sei f eine ganz transzendente Funktion. Eine komplexe Zahl w heißt kritischer Wert der Funktion f, falls es eine kpomplexe Zahl z mit f'(z)=0 und f(z)=w gibt. Eine komplexe Zahl b heißt asymptotischer Wert von f, falls es eine Kurve \gamma mit \gamma(t)\to\infty für t\to\infty gibt, so dass f(\gamma(t))\to b gilt. Die Menge sing(f^{-1}) ist die Menge der Singularitäten der Umkehrfunktion von f und besteht aus allen kritischen und allen asymptotischen Werten von f. Die Menge B aller ganz transzendenter Funktionen derart, dass sing(f^{-1}) beschränkt ist, heißt Eremenko-Lyubich-Klasse. Die Speiser-Klasse S besteht aus allen Funktionen der Klasse B, deren Menge der Singularitäten der Umkehrfunktion sogar endlich ist.
Insbesondere in der komplexen Dynamik, die sich mit dem Verhalten einer ganzen oder rationalen Funktion unter Iteration befasst, wurden diese Funktionenklassen ausgiebig untersucht. Die Konstruktion von Funktionen in den Klassen B und S mit vorgeschriebenem Verhalten ist von besonderem Interesse. Eine Möglichkeit, Funktionen der Klasse B zu konstruieren, sind sogenannte Cauchyintegrale. Gwyneth Stallard nutzte diese Methode, um zu beweisen, dass es für jedes d\in(1,2) eine Funktion in der Klasse B gibt, deren Juliamenge Hausdorff-Dimension d hat. Die Form der Trakte ihrer Funktionen bestimmt dabei die Hausdorff-Dimension. Dabei ist ein Trakt eine Zusammenhangskomponente der Menge, auf welcher der Absolutbetrag der Funktion groß ist.
Es ergibt sich die Frage, ob Stallards Resultat auch für Funktionen der Klasse S gilt. Auch wenn wir diese Frage nicht beantworten können, so zeigen wir, dass es Funktionen in der Klasse S gibt, deren Trakte in gewissem Sinne den von Stallard genutzten Trakten ähneln. Im Allgemeinen sind Funktionen, die mithilfe von Cauchyintegralen konstruiert wurden, nicht in der Klasse S. Eine alternative Konstruktionsmethode ist die quasikonforme Faltung, die kürzlich von Christopher Bishop vorgestellt wurde.
Wir nutzen Bishops Methode, um quasireguläre Funktionen zu konstruieren, die nur in einem parabelförmigen, zur reellen Achse symmetrischen Trakt wachsen und ansonsten beschränkt sind. Des Weiteren beweisen wir, dass es zu jeder so konstruierten quasiregulären Funktion g eine ganze Funktion f in der Klasse S gibt, so dass g=f\circ\phi für eine quasikonforme Abbildung \phi gilt. Somit ist der Trakt von f, welcher ebenfalls symmetrisch zur reellen Achse ist, ein quasikonformes Bild des Traktes von g. Ferner ist die hierbei genutzte quasikonforme Abbildung asymptotisch konform bei unendlich. Wir nutzen dieses, um zu zeigen, dass der Maximalbetrag M(r,f) von f auf dem Kreis mit Radius r von unten durch eine Funktion beschränkt ist, die von der Form des Traktes abhängt. Insbesondere beweisen wir, dass es eine ganze Funktion f in der Klasse S gibt, die nur einen Trakt hat, der ferner symmetrisch zur reellen Achse ist, so dass \log\log M(r,f) von unten durch d\cdot\sqrt{r} für ein d>0 beschränkt ist
LANDAU’S THEOREM FOR HOLOMORPHIC CURVES IN PROJECTIVE SPACE AND THE KOBAYASHI METRIC ON HYPERPLANE COMPLEMENTS
No full textAbstract. We prove an effective version of a theorem of Dufresnoy: For any set of 2n+1 hyperplanes in general position in P n, we find an explicit constant K such that for every holomorphic map f from the unit disc to the complement of these hyperplanes, we have f # (0) ≤ K, where f # denotes the norm of the derivative measured with respect to the Fubini-Study metric. This result gives an explicit lower bound on the Royden function, i.e., the ratio of the Kobayashi metric on the hyperplane complement to the Fubini-Study metric. Our estimate is based on the potential-theoretic method of Eremenko and Sodin. 1
Curvature and hyperbolicity of surfaces
No full textAn Aleksandrov surface is a generalization of two-dimensional Riemannian manifolds, and it is known by a theorem of A. Huber (1960) that every open simply connected Aleksandrov surface is conformally equivalent either to the unit disc (hyperbolic case) or to the plane (parabolic case). Thus one can study complex analysis on Aleksandrov surfaces, and in the first part of this thesis we prove a criterion for hyperbolicity of an Aleksandrov surface which has a nice tiling (or triangulation) and for which the negative curvature dominates. We apply this result to generalize a theorem of R. Nevanlinna and prove that a Riemann surface of class S is hyperbolic if negative excesses are spread uniformly over the corresponding Speiser graph. A partial answer for R. Nevanlinna\u27s conjecture about Speiser graphs follows. In the second part of this thesis, we study the relations between linear isoperimetric inequalities and Gromov hyperbolicity on Speiser graphs, their duals, and the corresponding Riemann surfaces of class S, and show the following results: if a linear isoperimetric inequality holds for one of these three spaces, so does for the others; Gromov hyperbolicity of a Riemann surface of class S is equivalent to that of the corresponding dual Speiser graph; a linear isoperimetric inequality on a dual Speiser graph or the corresponding Riemann surface of class S implies Gromov hyperbolicity of them. We also construct some counterexamples so as to disprove the other implications
Brody curves omitting hyperplanes
No full textA Brody curve, a.k.a. normal curve, is a holomorphic map f from the complex line C to the complex projective space Pn such that the family of its translations {z 7 → f(z+a) : a ∈ C} is normal. We prove that Brody curves omitting n hyperplanes in general position have growth order at most one, normal type. This generalizes a result of Clunie and Hayman who proved it for n = 1
Correction to the paper “On the second main theorem of Cartan”
No full textexponents sk in (20) can be complex, and this affects most of the arguments that follow. Below is the modified proof of Theorem 2. To prove Theorem 2, we use the following two facts about the class F: 1. F is a differential ring [2]. This means that F is closed under addition, multiplication and differentiation. 2. All functions y ∈ F are entire functions of completely regular growth in the sense of Levin–Pflüger [4], with piecewise-trigonometric indicators, the notions which we recall now. Let f be a holomorphic function in an angular sector S = {reiθ: |θ−θ0 | < ǫ, r> 0}. We say that f has completely regular growth with respect to order ρ> 0 if the following finite limit exists lim r→∞, reiθ 6∈E log |f(reiθ)| |r|
On the Second Main Theorem of Cartan
No full textThe possibility of reversion of the inequality in the Second Main Theorem of Cartan in the theory of holomorphic curves in projective space is discussed. A new version of this theorem is proved that be-comes an asymptotic equality for a class of holomorphic curves defined by solutions of linear differential equations
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