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 tip 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 naps to sufficiently small neighbourhoods of critical values have their higher order Schwarzian derivatives positive tip to any given order.
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