Higher order Schwarzian derivatives in interval dynamics

TL;DR

This paper introduces higher order Schwarzian derivatives in interval dynamics, generalizing classical results to control derivatives of high order for multimodal maps, with implications for the behavior near critical values.

Contribution

It develops a theory of higher order Schwarzian derivatives and extends the Koebe lemma to maps with positive higher order Schwarzian derivatives, enhancing understanding of interval map dynamics.

Findings

Inverse branches of first return maps have positive higher order Schwarzian derivatives near critical values.

Generalization of the Koebe lemma to higher order Schwarzian derivatives.

Control over derivatives of high order in multimodal interval maps.

Abstract

We introduce an infinite sequence of higher order Schwarzian derivatives closely related to the theory of monotone matrix functions. We generalize the classical Koebe lemma to maps with positive Schwarzian derivatives up to some order, obtaining control over derivatives of high order. For a large class of multimodal interval maps we show that all inverse branches of first return maps to sufficiently small neighbourhoods of critical values have their higher order Schwarzian derivatives positive up to any given order.