Some transcendental entire functions with irrationally indifferent fixed points

  • Kisaka Masashi
    Department of Mathematical Sciences, Graduate School of Human and Environmental Studies, Kyoto University
  • Naba Hiroto
    Department of Mathematical Sciences, Graduate School of Human and Environmental Studies, Kyoto University

抄録

<p>Let S be the set of all transcendental entire functions of the form P(z) exp(Q(z)), where P and Q are polynomials. In this paper, by using the theory of polynomial-like mappings, we construct various kinds of functions in S with irrationally indifferent fixed points as follows:</p><p>(1) We construct functions in S with bounded type Siegel disks centered at points other than the origin bounded by quasicircles containing critical points. This is an extension of Zakeri's result in [24] for fS.</p><p>(2) We construct functions in S with Cremer points whose multipliers satisfy some Cremer's condition in [6] only for rational functions. Our method shows that this condition can be applicable even in some transcendental cases.</p><p>(3) For any integer d ≥ 2 and some cC \ {0}, we show that the function of the form e2πiθz(1 + cz)d−1ez (θR\Q) has a Siegel point at the origin if and only if θ is a Brjuno number. This is an extension of Geyer's result in [11].</p><p>(4) For the function of the form (e2πiθz + αz2)ez (θR\Q, αC\{0}), we show that if α and θ satisfy some condition, then the Siegel disk centered at the origin is bounded by a Jordan curve containing a critical point, which is not a quasicircle. Moreover, we can choose α and θ so that the Lebesgue measure of the Julia set is positive and can also choose them so that it is zero. This is an extension of Keen and Zhang's result in [13].</p>

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詳細情報 詳細情報について

  • CRID
    1390857226420761344
  • DOI
    10.2996/kmj45304
  • ISSN
    18815472
    03865991
  • 本文言語コード
    en
  • データソース種別
    • JaLC
    • Crossref
    • KAKEN
  • 抄録ライセンスフラグ
    使用不可

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