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 seem somewhat arbitrary, as it is not inherently a dynamical condition, except for the fact that it is preserved under iteration. In this brief work, we show that the negative Schwarzian derivative condition is not arbitrary in any sense but is instead strictly related to the fulfillment of the Minimum Principle for the derivative of the map and its iterates, which plays a key role in the proof of Singer's Theorem.