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

TL;DR

This paper explores the significance of the negative Schwarzian derivative in one-dimensional dynamics, linking it to the Minimum Principle and providing a deeper understanding of its role in dynamical properties.

Contribution

It demonstrates that the negative Schwarzian derivative condition is fundamentally connected to the Minimum Principle, clarifying its importance in dynamical analysis.

Findings

Negative Schwarzian derivative relates to the Minimum Principle.

The condition is preserved under iteration.

Supports proof of Singer's Theorem.

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.