International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 29, Pages 1543-1549
doi:10.1155/S0161171204306253
Faà di Bruno's formula and nonhyperbolic fixed points of one-dimensional maps
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.
Abstract
Fixed-point theory of one-dimensional maps of ℝ
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.