Crinkly curves, Markov partitions and dimension

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Abstract

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.

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