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