# Invariants in fractal geometry

## Main Question or Discussion Point

Hello,

after reading something about fractals, I was wondering if it is possible to find invariants on fractal entities. For example in 3D Euclidean space we know that curvature and torsion uniquely define a regular curve: they are invariant to rigid motions.

In fractal geometry and in several papers dealing with fractal quantities, it seems to me that the only quantity that is invoked to "describe" a fractal, is its dimension (for example its http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension" [Broken]).

Is that really the only way to "distinguish" a fractal curve from another fractal curve?

Thanks.

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I haven't personally come across any others- except other dimensions such as the box counting dimension. Of course, you can always view the fractal as a topological/metric space and use invariants from that theory. But I'm sure there must be more invariants specific to fractal theory that exist.

Thanks!

Indeed, from what I understood from the books I am reading, the only invariants that are known and currently used are the various definitions of dimensions (as you said), and lacunarity. These are both invariant to bi-Lipschitz transformations, but they are far from representing complete invariants => different fractals can have same dimension and lacunarity.

I've not heard of lacunarity, I'll look it up.

You say "different fractals can have same dimension and lacunarity" but then, what do you mean by different fractals? That there isn't a bi-Lipschitz transformation between them?

I suppose the problem with fractals is that there isn't an extremely explicit way of defining what one is: in topology, there is a simple definition of what a topological space is, in metric spaces, what a metric space is, in group theory, what a group is, and it is very easy to define what the appropriate maps are that preserve their structures. What can we do for fractals? It's hard to think of what can be done, except to consider them as a subclass of metric spaces (topological spaces won't capture everything we want to know e.g. different Cantor sets can have different box counting dimensions). But once you've done this, you need to think of invariants which might apply to any metric space but capture the notion of your object being a fractal, which seems to be a very local thing, like dimension.

I think the Hausdorff dimension is an invariant of a fractal.

My bad; unfortunately, the "edit" function is disabled. Just wanted to state that
Mandelbrot , who defined a fractal curve as having
a Hausdorff dimension strictly larger than their topological dimension.

Oh, ok, thanks. The problems I highlighted still apply though. For example, when are two fractals "the same" or "different"? Do you need a bi-Lipschitz map? How then is the Hausdorff dimension an invariant?

Good question; I don't know if having a Bi-Lipschitz (don't ask, don't tell !) map
is enough and/or if Bi-Lipschitz preserves Hausdorff dimension. I'll look it up.

EDIT.
I imagine we also need to have the obvious homeomorphism, but I don't know
if Hausdorff dimension is a topological property; it may be a nice exercise to figure
that out.

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I am reasonably certain that a bi-Lipschitz map doesn't have to preserve the Hausdorff dimension.

I suppose fractal geometry is one of those areas where its not that helpful to have maps between fractals. I can imagine that the only class of maps to leave invariants that you want preserved alone are very strict (probably distance preserving).

The OP talks of invariance, but that is a very ambiguous word, there are lots of ways to categorise and quantify fractals, depending on what properties you want.
For example, the size of a fractal, the number of children at each branch (in the case of a tree), ordered vs random, how directed it is (whether children add to the leaves or all over), the depth of the fractal, pure fractal vs multi-fractal, self-crossing vs not, ... you name it.

Bacle2
I am reasonably certain that a bi-Lipschitz map doesn't have to preserve the Hausdorff dimension.

(probably distance preserving).
Jamma, what I meant was that I wonder if Lipchitz mapz that preserve Hausdorff dimension also preserve the fractal properties. It would be interesting to see if there is a category of fractals, and, if so, what the morphisms are,

TMFKAB:
The Mathematician* Formerly Known as Bacle **

* In training.

** Before my computer crashed and wiped my accounts and everything else.

Ah, I think I see what you mean. The Hausdorff dimension is the property of the spaces, not the maps though- would you say that the image of the maps must have the same Hausdorff dimension? Or possibly you could have a local definition of dimension and say that the local dimension of a point must be preserved under a map.

You could certainly define these categories, but how useful that'd be I'm not sure.

Pi and e are hidden in the structure of the mandelbrot set.

https://home.comcast.net/~davejanelle/mandel.html

I don't understand the proof in the PDF though, if anyone can explain, I would be grateful.

I'm also curious in learning more about fractals. I think Gaston Julia and Pierre Fatou's paper on iterated functions is probably a good start to understand the basics, but it doesn't explain why certain structures arise to be the way they are.

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