Fractal Geometry

I did a forth year project on fractal geometry. The main purpose of the project was to investigate the idea of Hausdorff Dimension.
The idea arises from measure theory, introduced in section 3.
In section 4 the Hausdorff Dimension is introduced. While doing so we prove the first main result of this project. It is a very technical but important result which insures that Hausdorff measure is in fact a measure on the sets we want to investigate.
The main idea of the Hausdorff dimension is the ability to attach a unique dimension to fractal sets, which usually don’t seem to have integer dimension. Hausdorff dimension is a generalization of the concept of dimension to non-integers. Clearly it would not be of much use if it did not attach integers to the non-fractal sets as we are used to. This leads us to our second main result which states that the Hausdorff measure is equal to the Lebesgue measure up to a constant factor on Euclidean subsets of full dimension. We prove this in section 4.
The generality of Hausdorff dimension often makes it very difficult to calculate. Often it shows rewarding to introduce more specific definitions of dimension which gains transparency in exchange for generality. Sometimes we can prove that these simplifications are actually equal to the Hausdorff dimension on certain sets. In section 5 we introduce the box-dimension which is a useful simplification of the idea of the Hausdorff dimension.
As an application of the concepts introduced we turn our attention to iterated function systems (IFS) in section 6. An IFS is a system of functions contracting the sets they are applied to. We prove the third main result, which states that an IFS has a unique attractor, that is, there is a unique set which is invariant under iterations through the whole of the IFS. Any element in our collection of sets will tend to the attractor, therefore the name. And further, stated as a separate theorem, if the IFS contracts sets by well defined ratios the attractor has finite non-zero Hausdorff dimension which is equal to the box-dimension of the set. The last theorem even gives a useful method of determining the dimension of a set.
In section 7 we apply the theory developed to some fractals and show a technique for finding the IFS, given a fractal like the one on the cover.
We conclude the project by asking ourselves whether the Hausdorff dimension is really the preferable definition of dimension. Over the resent years some have pointed out another definition, namely the Minkowski dimension. In section 8 we look at the differences between the Minkowski and Hausdorff dimensions. The Minkowski dimension is closely related to the box-dimension and thus easier to calculate. It has some unpleasant properties seen from the theoretical point of view, but these properties makes it able to extract information which is out of reach of the Hausdorff dimension. This has wide applications in inverse spectral problems and makes it a more preferable dimension for applications.

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