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. Dierent 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 similitudes, such sets are constructed by an

iterative procedure using an initial" and a standard" polygon. However, here we

need to consider instead the set S.

It turns out, somewhat surprisingly at rst, that the invariant set K is deter-

mined by S. In fact, for given S there exists a unique compact set K invariant

with respect to S. Furthermore, K is the limit of various approximating sequences

of sets which can be constructed from S.

More precisely we have the following result from 3.1(3), 3.2.

(1) Let X = (X; d) be a complete metric space and S = fS1; : : : ; SNg be a

nite set of contraction maps (2.2) on X. Then there exists a unique closed bounded

set K such that K = SN

i=1 SiK. Furthermore, K is compact and is the closure of

the set of xed points si1:::ip of nite compositions Si1 


 Sip of members of S.

For arbitrary A  X let S(A) = SN

i=1 SiA, Sp(A) = S(Sp��1(A)). Then for

closed bounded A, Sp(A) ! K in the Hausdor metric (2.4).

The compact set K in (1) is denoted jSj. jSj supports various measures in a

natural way. We have the following from 4.4.

(2) In addition to the hypotheses of (1), suppose 1; : : : ; N 2 (0; 1) and

PN

i=1 i = 1. Then there exists a unique Borel regular measure  of total mass 1

such that  = PN

i=1 iSi#(). Furthermore spt() = jSj.

The measure  is denoted kS; k.

The set jSj will not normally have integral Hausdor dimension. However, in

case (X; d) is Rn with the Euclidean metric, jSj can often be treated as an m-

dimensional object, m an integer, in the sense that there is a notion of integration

of C1 m-forms over jSj. In the language of geometric measure theory (2.7), jSj supports an m-dimensional integral

at chain. The main result here is 6.3(3).

Now suppose (X; d) is Rn with the Euclidean metric, and the Si 2 S are simil-

itudes. Let Lip Si = ri (2.2) and let