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Crinkly curves, Markov partitions and dimension
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Bedford, T. (1984) Crinkly curves, Markov partitions and dimension. PhD thesis, University of Warwick.

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Official URL: http://webcat.warwick.ac.uk/record=b1464305~S1
Abstract
We consider the relationship between fractals and
dynamical systems. In particular we look at how the
construction of fractals in (D1) can be interpretedin 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 nondense 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 arenot similitudes. Examples
are given for which capacity and Hausdorff dimension give
different answers.
Item Type:  Thesis or Dissertation (PhD) 

Subjects:  Q Science > QA Mathematics 
Library of Congress Subject Headings (LCSH):  Fractals, Global analysis (Mathematics), Partitions (Mathematics) 
Official Date:  September 1984 
Institution:  University of Warwick 
Theses Department:  Mathematics Institute 
Thesis Type:  PhD 
Publication Status:  Unpublished 
Supervisor(s)/Advisor:  Series, Caroline 
Sponsors:  Science and Engineering Research Council (Great Britain) (SERC) 
Extent:  vii, 110 p. 
Language:  eng 
URI:  http://wrap.warwick.ac.uk/id/eprint/50539 
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