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

Vadim Ponomarenko

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

Received 22 June 2003

Abstract

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.