Fatou’s Associates

Q3 Mathematics
Vasiliki Evdoridou, Lasse Rempe, David J. Sixsmith
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引用次数: 3

Abstract

Suppose that f is a transcendental entire function, \(V \subsetneq {\mathbb {C}}\) is a simply connected domain, and U is a connected component of \(f^{-1}(V)\). Using Riemann maps, we associate the map \(f :U \rightarrow V\) to an inner function \(g :{\mathbb {D}}\rightarrow {\mathbb {D}}\). It is straightforward to see that g is either a finite Blaschke product, or, with an appropriate normalisation, can be taken to be an infinite Blaschke product. We show that when the singular values of f in V lie away from the boundary, there is a strong relationship between singularities of g and accesses to infinity in U. In the case where U is a forward-invariant Fatou component of f, this leads to a very significant generalisation of earlier results on the number of singularities of the map g. If U is a forward-invariant Fatou component of f there are currently very few examples where the relationship between the pair (fU) and the function g has been calculated. We study this relationship for several well-known families of transcendental entire functions. It is also natural to ask which finite Blaschke products can arise in this manner, and we show the following: for every finite Blaschke product g whose Julia set coincides with the unit circle, there exists a transcendental entire function f with an invariant Fatou component such that g is associated with f in the above sense. Furthermore, there exists a single transcendental entire function f with the property that any finite Blaschke product can be arbitrarily closely approximated by an inner function associated with the restriction of f to a wandering domain.

Fatou的合伙人
假设f是超越整函数,\(V\subsetneq{\mathbb{C}})是单连通域,U是\(f^{-1}(V)\)的连通分量。使用黎曼映射,我们将映射\(f:U\rightarrow V\)与内函数\(g:{\mathbb{D}}\rightarrow{\math bb{D})相关联。很容易看出,g要么是有限Blaschke乘积,要么通过适当的归一化,可以被视为无限Blaschke积。我们证明,当f在V中的奇异值远离边界时,g的奇异性与U中无穷大的可达性之间存在很强的关系。在U是f的前向不变Fatou分量的情况下,这导致了关于映射g的奇异数的早期结果的非常显著的推广。如果U是f的前向不变Fatou分量,则目前很少有计算对(f,U)和函数g之间关系的例子。我们研究了几个著名的超验整体函数族的这种关系。同样自然地,我们会问哪些有限Blaschke乘积可以以这种方式产生,我们展示了以下内容:对于每一个Julia集与单位圆重合的有限Blaschke-乘积g,都存在一个具有不变Fatou分量的超越整体函数f,使得g在上述意义上与f相关联。此外,存在一个单一的超越整体函数f,其性质是任何有限的Blaschke乘积都可以由与f在游荡域的限制相关的内函数任意逼近。
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来源期刊
Arnold Mathematical Journal
Arnold Mathematical Journal Mathematics-Mathematics (all)
CiteScore
1.50
自引率
0.00%
发文量
28
期刊介绍: The Arnold Mathematical Journal publishes interesting and understandable results in all areas of mathematics. The name of the journal is not only a dedication to the memory of Vladimir Arnold (1937 – 2010), one of the most influential mathematicians of the 20th century, but also a declaration that the journal should serve to maintain and promote the scientific style characteristic for Arnold''s best mathematical works. Features of AMJ publications include: Popularity. The journal articles should be accessible to a very wide community of mathematicians. Not only formal definitions necessary for the understanding must be provided but also informal motivations even if the latter are well-known to the experts in the field. Interdisciplinary and multidisciplinary mathematics. AMJ publishes research expositions that connect different mathematical subjects. Connections that are useful in both ways are of particular importance. Multidisciplinary research (even if the disciplines all belong to pure mathematics) is generally hard to evaluate, for this reason, this kind of research is often under-represented in specialized mathematical journals. AMJ will try to compensate for this.Problems, objectives, work in progress. Most scholarly publications present results of a research project in their “final'' form, in which all posed questions are answered. Some open questions and conjectures may be even mentioned, but the very process of mathematical discovery remains hidden. Following Arnold, publications in AMJ will try to unhide this process and made it public by encouraging the authors to include informal discussion of their motivation, possibly unsuccessful lines of attack, experimental data and close by research directions. AMJ publishes well-motivated research problems on a regular basis.  Problems do not need to be original; an old problem with a new and exciting motivation is worth re-stating. Following Arnold''s principle, a general formulation is less desirable than the simplest partial case that is still unknown.Being interesting. The most important requirement is that the article be interesting. It does not have to be limited by original research contributions of the author; however, the author''s responsibility is to carefully acknowledge the authorship of all results. Neither does the article need to consist entirely of formal and rigorous arguments. It can contain parts, in which an informal author''s understanding of the overall picture is presented; however, these parts must be clearly indicated.
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