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- W1520661390 abstract "We consider the relationship between fractals anddynamical systems. In particular we look at how theconstruction of fractals in (D1) can be interpreted-in adynamical setting and additionally used as a simple methodof describing the construction of invariant sets ofdynamical systems. There is often a confusion betweenHausdorff dimension and capacity -which is much easier tocompute- and we show that simple examples of fractals,arising in dynamical systems, exist for which the twoquantities differ.In Chapter One we outline the mathematical backgroundrequired 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 howto interpret this construction dynamically. This approachenables us to calculate Hausdorff dimension and describeHausdorff measure for certain recurrent sets. We alsoprove a conjecture of Dekking about conditions under whichthe best general estimate of dimension actually equalsdimension.In Section One of Chapter Three recurrent sets are usedto construct special Markou partitions for expandingendomorphisms of T2 and hyperbolic automorphisms of T3.These partitions have transition matrices closely relatedto the covering maps. It is also shown that Markovpartitions can be constructed for the same map whoseboundaries have different capacities. Section Two looksat the problem of coding between two Markov partitionsfor the same expanding endomorphism of T2. It is shownthat there is a relationship between mean coding time andthe capacities of the boundaries. Section Three usesrecurrent sets to construct fractal subsets of toriwhich have non-dense orbits under the above mappings.Finally, Chapter Four calculates capacity and Hausdorffdimension for a class of fractals (which are also recurrentsets) whose scaling maps are-not similitudes. Examplesare given for which capacity and Hausdorff dimension givedifferent answers." @default.
- W1520661390 created "2016-06-24" @default.
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- W1520661390 date "1984-09-01" @default.
- W1520661390 modified "2023-09-26" @default.
- W1520661390 title "Crinkly curves, Markov partitions and dimension" @default.
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