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Conformal Geometry and Dynamics
Conformal Geometry and Dynamics
ISSN 1088-4173

 

The Schwarzian operator: sequences, fixed points and $ N$-cycles


Author: Stephen M. Zemyan
Journal: Conform. Geom. Dyn. 15 (2011), 44-49
MSC (2010): Primary 34L30; Secondary 30D30, 34A25, 34A34
Published electronically: April 25, 2011
MathSciNet review: 2801171
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Abstract | References | Similar Articles | Additional Information

Abstract: 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.


References [Enhancements On Off] (What's this?)

  • 1. Lars V. Ahlfors, Complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable; International Series in Pure and Applied Mathematics. MR 510197 (80c:30001)

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Additional Information

Stephen M. Zemyan
Affiliation: Department of Mathematics, Penn State Mont Alto, Mont Alto, Pennsylvania 17237-9799
Email: smz3@psu.edu

DOI: http://dx.doi.org/10.1090/S1088-4173-2011-00224-4
PII: S 1088-4173(2011)00224-4
Keywords: Schwarzian derivative
Received by editor(s): December 9, 2010
Published electronically: April 25, 2011
Article copyright: © Copyright 2011 American Mathematical Society