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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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The Schwarzian operator: sequences, fixed points and $N$-cycles
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by Stephen M. Zemyan PDF
Conform. Geom. Dyn. 15 (2011), 44-49 Request permission


Given a function $f(z)$ that is analytic in a domain $D$, we define the classical Schwarzian derivative $\{f,z\}$ of $f(z)$, and mention some of its most useful analytic properties. We explain how the process of iterating the Schwarzian operator produces a sequence of Schwarzian derivatives, and we illustrate this process with examples. Under a suitable restriction, these sequences become $N$-cycles of Schwarzian derivatives. Some properties of functions belonging to an $N$-cycle are listed. We conclude the article with a collection of related open problems.
  • Lars V. Ahlfors, Complex analysis, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable. MR 510197
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Additional Information
  • Stephen M. Zemyan
  • Affiliation: Department of Mathematics, Penn State Mont Alto, Mont Alto, Pennsylvania 17237-9799
  • Email:
  • Received by editor(s): December 9, 2010
  • Published electronically: April 25, 2011
  • © Copyright 2011 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 15 (2011), 44-49
  • MSC (2010): Primary 34L30; Secondary 30D30, 34A25, 34A34
  • DOI:
  • MathSciNet review: 2801171