TY - JOUR

T1 - The schwarzian operator

T2 - Sequences, fixed points and N-cycles

AU - Zemyan, Stephen M.

PY - 2011/4/25

Y1 - 2011/4/25

N2 - 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.

AB - 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.

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U2 - 10.1090/S1088-4173-2011-00224-4

DO - 10.1090/S1088-4173-2011-00224-4

M3 - Article

AN - SCOPUS:85009826999

VL - 15

SP - 44

EP - 49

JO - Conformal Geometry and Dynamics

JF - Conformal Geometry and Dynamics

SN - 1088-4173

IS - 4

ER -