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

Alec Norton将研究与二维可微映射的动力学行为有关的问题。这些又与关于可微函数的临界集的性质的问题密切相关。例如，关于一个临界集的图像大小的信息可以从有关函数的可微度中得到。诸如此类的结果与莫尔斯-萨德定理密切相关。这些研究在许多方面是对更成熟的保角动力学工作的补充。尽管莫尔斯-萨德已被广泛用作分析工具，诺顿将研究其几何分支。特别是他将用它来研究光形映射问题。这要求二维球面的微分同胚，其极点满足某些性质。以往的研究表明，函数的可微度在这些问题中起着重要的作用。在一个稍微不同的方向上，他将尝试将关于圆上Denjoy动力学的已知结果扩展到环面。

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.