5

I am reading several books on Fractals and their geometry. Pretty much all of them say "Hutchinson showed you can use the self-similarity property of fractals to help calculate their Hausdorff dimension". None of them, however, give a proof - not even a sketch of one - in their books (in fact, they don't even state it formally). I was hoping someone could (a) show me how to prove it, or (b) explain to me that it is far too difficult to bother with.

Theorem (how I think it should be stated formally). Let $F\subset\mathbb{R}^{m}$ and let $K_{1},\ldots, K_{m}$ be contractions on $\mathbb{R}^{m}$ with Lipschitz constants $c_{1},\ldots,c_{m}$, and let $c=\max\{c_{1},\ldots,c_{m}\}$. Define the Hutchinson operator $K$ on $\mathbb{R}^{m}$ by $K(F)=K_{1}(F)\cup\ldots\cup K_{m}(F)$. Then $K$ is a contraction with Lipschitz constant $c$.

Proof. Not a clue.

user458984
  • 325
  • Take for example the Cantor set $C$ (initial: given an interval, next step: erase the middle third of all the intervals you see). If we zoom it by a radius of $3$, it is exactly a copy of $2$ Cantor sets of the original size. In the end, its dimension will be $\log_32$. – Berci Jul 26 '17 at 00:03
  • In the context of Hutchinson's thm, you can take two contractions to illustrate the same: $x\mapsto x/3$ and $x\mapsto (x+2)/3$ if we start out from $[0,1]$. – Berci Jul 26 '17 at 00:09
  • 1
    I guess I wonder which books on Fractal Geometry you've been reading? :) Self-similarity is covered in detail in Chapter 9 of Falconer's Fractal Geometry. The theorem that you formulate (which is not immediately related to the computation of dimension) is part of his theorem 9.1. Techniques for computing dimension based on this are covered later in the chapter. – Mark McClure Jul 26 '17 at 01:32

1 Answers1

1

Note that Lipschitz mappings are probably not good enough in this context---your quoted text requires that the mappings be self-similar, which is a stronger condition. "Hutchinson's theorem," which deals with self-similar sets, is proved in the paper

Hutchinson, John E. "Fractals and self similarity," Indiana Mathematics Journal, 30(5) (Sept-Oct 1981).

This paper is quite approachable, and sets up much of the modern theory of fractal analysis, and the proofs are not that hard to follow. I really wish that someone had brought it to my attention when I was first learning about the Hausdorff measure of self-similar sets.

Xander Henderson
  • 29,772
  • Thank you for the information, it does look very nice. Do you know on which page the proof is? I have had a look and cannot find it. Thanks. – user458984 Aug 13 '17 at 13:40
  • @user458984 I am not exactly sure what theorem you are trying to prove, but I am fairly certain that the result alluded to in your original question is iterated functions systems satisfying the open set condition have Hausdorff dimension equal to their similarity dimension. This starts around page 735. – Xander Henderson Aug 13 '17 at 14:53