1,355,740 research outputs found
The Eremenko-Lyubich constant
No full textEremenko and Lyubich proved that an entire function whose set of singular
values is bounded is expanding at points where its image has large modulus.
These expansion properties have been at the centre of the subsequent study of
this class of functions, now called the Eremenko-Lyubich class. We improve the
estimate of Eremenko and Lyubich, and show that the new estimate is
asymptotically optimal. As a corollary, we obtain an elementary proof that
functions in the Eremenko-Lyubich class have lower order at least
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
A new characterisation of the Eremenko-Lyubich class
No full textThe Eremenko-Lyubich class of transcendental entire functions with a bounded set of singular values has been much studied. We give a new characterisation of this class of functions. We also give a new result regarding direct singularities which are not logarithmic
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
From pathological to paradigmatic: A retrospective on Eremenko and Lyubich's entire functions
No full textThis paper surveys the impact of Eremenko and Lyubich’s paper “Examples of entire functions with pathological dynamics”, published in 1987 in the Journal of the London Mathematical Society. Through a clever extension and use of classical approximation theorems, the authors constructed examples exhibiting behaviours previously unseen in holomorphic dynamics. Their work laid foundational techniques and posed questions that have since guided a good part of the development of transcendental dynamics
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
A Newly Compiled Checklist of the Vascular Plants of the Habomais, the Little Kurils
No full textThe new floristic checklist of the Habomais, the Little Kurils, was compiled from Barkalov and Eremenko (2003) and Eremenko (2003), and supplemented by the specimens collected by Gage and Joneson in 1998 and Eremenko in 2002. In the checklist, 61 families, 209 genera and 332 species were recognized. Scientific and vernacular names commonly adopted in Russian and Japanese taxonomic references are listed and compared, and some taxonomic notes are also added. This list will contribute the future critical taxonomic and nomenclatural studies on the vascular plants in this region. The plants of each individual island in the Habomais are listed in the table.Biodiversity and Biogeography of the Kuril Islands and Sakhalin vol.
On a question of Eremenko concerning escaping components of entire functions
No full textLet f be an entire function with a bounded set of singular values, and suppose furthermore that the postsingular set of f is bounded. We show that every component of the escaping set I(f) is unbounded. This provides a partial answer to a question of Eremenko
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
Several Constructions in the Eremenko-Lyubich Class
No full textWe use a theorem of Bishop to construct several functions in the Eremenko-Lyubich class . First it is verified, that in Bishop\u27s initial construction of a wandering domain in , all wandering Fatou components must be bounded. Next we modify this construction to produce a function in with wandering domain and uncountable singular set. Finally we construct a function in with unbounded wandering Fatou components. It is shown that these constructions answer two questions posed by Osborne and Sixsmith. | 48 page
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