Fixed-point theory of one-dimensional maps of R does not completely address the issue of non-hyperbolic fixed points. This paper 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.
Ponomarenko, V. (2004). Faà di Bruno's formula and nonhyperbolic fixed points of one-dimensional maps. International Journal of Mathematics and Mathematical Sciences, 2004(29), 1543-1549. doi:10.1155/S0161171204306253
International Journal of Mathematics and Mathematical Sciences