One dimensional dynamics : cross-ratios, negative Schwarzian and structural stability
This thesis concerns the behaviour of maps with a unique critical point which is either a maximum or a minimum: so-called unimodal maps. Our first main result proves that for C2+η unimodal maps with non-flat critical point we have good control on the behaviour of cross-ratios on small scales. This result, an improvement on a result of Kozlovski in [K2], proves that in many cases the negative Schwarzian condition (which is not even defined if a map is not C3) is unnecessary. This result follows recent work of Shen, van Strien and Vargas. The main tools are standard cross-ratio estimates, the usual principal nest, the Koebe Lemma, the real bounds from [SV] and the 'Yoccoz Lemma'. Our second main result concerns questions of structural stability. Prompted by the final section of Kozlovski's thesis [K1], we prove that in some cases we can characterise those points at which a small local perturbation changes the type of the map. We prove for these cases that this set of 'structurally sensitive points' is precisely the postcritical set. The main tools are the Koebe Lemma, the real bounds of [LS1], and the quasiconformal deformation argument of [K3]. The thesis is arranged in the form of two chapters dealing with each of the main results separately, followed by an appendix to prove an auxiliary result. The chapters may be read independently of each other.