One dimensional dynamics : cross-ratios, negative Schwarzian and structural stability
Todd, Michael (2003) One dimensional dynamics : cross-ratios, negative Schwarzian and structural stability. PhD thesis, University of Warwick.
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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.
|Item Type:||Thesis or Dissertation (PhD)|
|Subjects:||Q Science > QA Mathematics|
|Library of Congress Subject Headings (LCSH):||Mappings (Mathematics), Critical point theory (Mathematical analysis)|
|Official Date:||September 2003|
|Institution:||University of Warwick|
|Theses Department:||Mathematics Institute|
|Sponsors:||Engineering and Physical Sciences Research Council (EPSRC) ; European Science Foundation (ESF)|
|Extent:||v, 95 p.|
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