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Fractals And Self Similarity

FRACTALS AND SELF SIMILARITY
Contents
1. Introduction 2
2. Preliminaries 3
2.1. Sequences of Integers 3
2.2. Maps in Metric Spaces 4
2.3. Similitudes 4
2.4. Hausdor Metric 6
2.5. Measures 6
2.6. Hausdor Measure 7
2.7. Geometric Measure Theory 8
3. Invariant Sets 10
3.1. Elementary Proof of Existence and Uniqueness, and Discussion of
Properties 10
3.2. Convergence in the Hausdor Metric 12
3.3. Examples 13
3.4. Remark 14
3.5. Parametrised Curves 14
4. Invariant Measures 15
4.1. Motivation 15
4.2. Denitions 16
4.3. The L metric 16
4.4. Existence and Uniqueness 16
4.5. Di erent Sets of Similitudes Generating the Same Set 17
5. Similitudes 18
5.1. Self-Similar Sets 18
5.2. Open Set Condition 18
5.3. Existence of Self Similar Sets. 19
5.4. Purely Unrectiable Sets. 21
5.5. Parameter Space 24
6. Integral Flat Chains 24
6.1. The F-metric. 24
6.2. The C-metric 25
6.3. Invariant Chains 26
References 27
1
2 JOHN E. HUTCHINSON
1. Introduction
Sets with non-integral Hausdor dimension (2.6) are called fractals by Mandel-
brot. Such sets, when they have the additional property of being in some sense
either strictly or statistically self-similar, have been used extensively by Mandel-
brot and others to model various physical phenomena (c.f. [MB] and the references
there). However, these notions have not so far been studied in a general framework.
In this paper we set up a theory of (strictly) self-similar objects, in a subsequent
paper we analyse statistical self-similarity.
We now proceed to indicate the main results. The reader should refer to the
examples in 3.3 for motivation. We say the compact set K  Rn is invariant if
there exists a nite set S = fS1; : : : ; SNg of contraction maps on K  Rn such
that
K =
N
[i=1
SiK:
In such a case we say K is invariant with respect to S. Often, but not always, the
Si will be similitudes, i.e. a composition of an isometry and a homothety (2.3).
In [MB], and in the case the Si are...