International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 29, Pages 1543-1549

Faà di Bruno's formula and nonhyperbolic fixed points of one-dimensional maps

Vadim Ponomarenko

Department of Mathematics, Trinity University, 715 Stadium Drive, San Antonio 78212-7200, TX, USA

Received 22 June 2003

Copyright © 2004 Vadim Ponomarenko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Fixed-point theory of one-dimensional maps of does not completely address the issue of nonhyperbolic fixed points. This note generalizes the existing tests to completely classify all such fixed points. To do this, a family of operators are exhibited that are analogous to generalizations of the Schwarzian derivative. In addition, a family of functions f are exhibited such that the Maclaurin series of f(f(x)) and x are identical.