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

亚历克·诺顿将致力于研究与二维可微映射的动力学行为有关的问题。这些反过来又与关于临界可微函数集的性质的问题密切相关。例如，关于临界集的图像大小的信息可以从有关函数的可微性程度中获得。这样的结果与Morse-Sard定理密切相关。这些研究在许多方面是对更成熟的共形动力学工作的补充。尽管Morse-Sard已被广泛用作一种分析工具，但诺顿仍将研究其几何影响。特别是，他将用它来研究斜行映射问题。这就要求二维球面的微分同胚，它的极点满足某些性质。以往的工作已经表明，函数的可微性程度在这些问题中起着重要的作用。在一个略有不同的方向上，他将尝试将圆上关于Denjoy动力学的已知结果推广到环面上。

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.