Crinkly curves, Markov partitions and dimension
We consider the relationship between fractals and dynamical systems. In particular we look at how the construction of fractals in (D1) can be interpreted-in a dynamical setting and additionally used as a simple method of describing the construction of invariant sets of dynamical systems. There is often a confusion between Hausdorff dimension and capacity -which is much easier to compute- and we show that simple examples of fractals, arising in dynamical systems, exist for which the two quantities differ. In Chapter One we outline the mathematical background required in the rest of the thesis. Chapter Two reviews the work of F. M. Dekking on generating 'recurrent sets', which are types of fractals. We show how to interpret this construction dynamically. This approach enables us to calculate Hausdorff dimension and describe Hausdorff measure for certain recurrent sets. We also prove a conjecture of Dekking about conditions under which the best general estimate of dimension actually equals dimension. In Section One of Chapter Three recurrent sets are used to construct special Markou partitions for expanding endomorphisms of T2 and hyperbolic automorphisms of T3. These partitions have transition matrices closely related to the covering maps. It is also shown that Markov partitions can be constructed for the same map whose boundaries have different capacities. Section Two looks at the problem of coding between two Markov partitions for the same expanding endomorphism of T2. It is shown that there is a relationship between mean coding time and the capacities of the boundaries. Section Three uses recurrent sets to construct fractal subsets of tori which have non-dense orbits under the above mappings. Finally, Chapter Four calculates capacity and Hausdorff dimension for a class of fractals (which are also recurrent sets) whose scaling maps are-not similitudes. Examples are given for which capacity and Hausdorff dimension give different answers.