On the Use of the Schwarzian derivative in Real One-Dimensional Dynamics

Abstract

In the study of properties within one-dimensional dynamics, the assumption of a negative Schwarzian derivative has been shown to be very useful. However, this condition may appear somewhat arbitrary, as it is not a dynamical condition in any sense other than that it is preserved for its iterates. In this brief work, we show that the assumption of a negative Schwarzian derivative it is not entirely arbitrary but rather strictly related to the fulfillment of the Minimum Principle for the derivative of the map and its iterates, which is the key point in the proof of Singer's Theorem.