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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Cauchy transforms of self-similar measures: Starlikeness and univalence
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by Xin-Han Dong, Ka-Sing Lau and Hai-Hua Wu PDF
Trans. Amer. Math. Soc. 369 (2017), 4817-4842 Request permission

Abstract:

For the contractive iterated function system $S_kz=e^{2\pi ik/m}+{\rho (z-e^{2\pi ik/m})}$ with $0<\rho <1, k=0,\cdots , m-1$, we let $K\subset \mathbb {C}$ be the attractor, and let $\mu$ be a self-similar measure defined by $\mu =\frac 1m\sum _{k=0}^{m-1}\mu \circ S_k^{-1}$. We consider the Cauchy transform $F$ of $\mu$. It is known that the image of $F$ at a small neighborhood of the boundary of $K$ has very rich fractal structure, which is coined the Cantor boundary behavior. In this paper, we investigate the behavior of $F$ away from $K$; it has nice geometry and analytic properties, such as univalence, starlikeness and convexity. We give a detailed investigation for those properties in the general situation as well as certain classical cases of self-similar measures.
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Additional Information
  • Xin-Han Dong
  • Affiliation: College of Mathematics and Econometrics, Hunan University, Changsha, 410082, People’s Republic of China – and – Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China
  • MR Author ID: 240828
  • Email: xhdong@hunnu.edu.cn
  • Ka-Sing Lau
  • Affiliation: College of Mathematics and Econometrics, Hunan University, Changsha, 410082, People’s Republic of China – and – Department of Mathematics, The Chinese University of Hong Kong, Hong Kong
  • MR Author ID: 190087
  • Email: kslau@math.cuhk.edu.hk
  • Hai-Hua Wu
  • Affiliation: College of Mathematics and Econometrics, Hunan University, Changsha, 410082, People’s Republic of China – and – Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China
  • Email: hunaniwa@163.com
  • Received by editor(s): December 12, 2014
  • Received by editor(s) in revised form: July 16, 2015
  • Published electronically: December 7, 2016
  • Additional Notes: This research was supported in part by the NNSF of China (No. 11571099), an HKRGC grant, SRFDP of Higher Education (No. 20134306110003), and Scientific Research Fund of Hunan Provincial Education Department (No. 14K057). The first author is the corresponding author
  • © Copyright 2016 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 369 (2017), 4817-4842
  • MSC (2010): Primary 28A80; Secondary 30C55, 30E20
  • DOI: https://doi.org/10.1090/tran/6819
  • MathSciNet review: 3632551