From dynamics to geometry on self-affine sets and measures

In school geometry, we learn to give names a number of simple geometric shapes, such as circles, lines and polygons. However, these shapes are of little help in accurate modeling of real shapes in nature: Clouds are not spheres, coastlines are not circles, nor does lightning draw straight lines. With purpose to describe these real, natural shapes, there exists a field of mathematics known as fractal geometry. The ultimate goal of the field is to create a unified language one can use to describe and compare such shapes in nature that we would otherwise have to describe as complex or formless. Such shapes are commonly referred to as fractals, and examples of such include coastlines, snowflakes, and lungs.

Since fractals do not bear resemblance to any shape of classical geometry, it is also often difficult or even impossible to measure their size with geometric quantities such as length, area or volume. For example, according to one source, the length of the coastline of Norway is approximately 50 000 kilometers, while another source claims the length to be over 100 000 kilometers: The more accurate a satellite image is used in the measurements, the longer length is measured for the coastline. Because of this, the size of a fractal is often more reasonable to measure using fractal dimension. By computing the fractal dimensions of two fractals which may look almost identical to the eye, it is possible to detect notable differences in their structures: For example, accoriding to some medical studies, human lungs having unusually small fractal dimension may indicate structural abnormalities caused by a disease.

In my doctoral dissertation, I study properties of fractals which are often essential to know from the point of view of applications. For example, it is often essential that the photo of a fractal always has the same fractal dimension regardless of the direction the photo has been taken from. Another important property is that a substantial part of the structure of a fractal cannot be destroyed by distortions which are random or irrelevant, in a sense. In my doctoral thesis, I confirm these and other similar properties for many fractals for which they were previously unknown. However, in my research I do not consider images of coastlines or lungs, but general mathematical models of fractals which can be simultaneously used to model many different fractals in nature.