Stretching and folding of the Lorenz attractor

Question

A common way to look at strange attractors (an example of which is the Lorenz attractor) is via a series of stretches and folds. I know the stretching of the Lorenz attractor is due to the presence of two negative Lyapunov exponents and one positive. I am, however, confused about the origin of the folding. I have read (in a source I can no longer find) that this is due to the contracting volume. This contraction of volume is however a consequence of the Lyapunov exponents and thus I can't see how this can cause the folding. My question is therefore in strange attractors (and specifically the Lorenz attractor) what causes the folding?

Answer 1

There are many definitions of attractor, but "folding" is neither a condition not a consequence in any of these definitions (to my best knowledge). Take for example a Julia set. There is no folding there since there is no room for it (in $\mathbb C$, while in $\mathbb C^2$ there are certainly examples).

Let me give some rough description for folding assuming that there is folding (actually, this is a rough criterion for folding no matter if we are talking about attractors or not):

If we have expansion and contraction on an open set along directions that don't change too much, and the ambient space is compact, the images (and preimages) under iteration of the open set will stretch a lot and because of compactness they will unavoidably bend causing the folding that you describe.

As a very simple example, take a Smale horseshoe. For a more elaborate example take the behavior caused by a tranverse homoclinic point.

Unfortunately, all is often much more complicated in "real life" examples such as in the Lorenz attractor (whose existence as you might know was only proved rigorously recently). In this particular case there is a similar mechanism to that in the Smale horseshoe but with "less uniform" expansion and contraction.