Mathematical Sciences: Fractal Geometry and the Dynamics of Finitely Differentiable Maps

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Alec Norton will work on problems related to the dynamical behaviour of differentiable mappings in two dimensions. These in turn are closely related to questions about the nature of the critical sets of differentiable functions. For example, information about the size of the image of a critical set can be obtained from the degree of differentiability of the function concerned. Results such as these are closely related to the Morse-Sard theorem. These investigations are in many ways complementary to the more established works on conformal dynamics. Although the Morse-Sard has been used extensively as an analytic tool, Norton will be investigating its geometric ramifications. In particular he will use it to study the loxodromic mapping problem. This asks for diffeomorphisms of the two dimensional sphere for which the poles satisfy certain properties. Previous work has shown that the degree of differentiability of the function plays an important role in these questions. In a slightly different direction he will try to extend known results about Denjoy dynamics on the circle to the torus.