Can a fractal be a manifold?

Question

Here it is said that it is not possible:

Can a fractal be a manifold? if so: will its boundary (if exists) be strictly one dimension lower?

But I am confused about this. What about the invariant manifolds (stable and unstable manifolds) of chaotic flows? Here it says they are fractals:

https://books.google.es/books?id=iERT8akgRUcC&pg=PA14&lpg=PA14&dq=fractal+unstable+manifold&source=bl&ots=o8eDVH3x8G&sig=cYEQ6R4XKGgH1EU2IQgIBlrpTl8&hl=es&sa=X&ved=0CDkQ6AEwAmoVChMIk_mUwIm9xwIVRFwUCh0c0AqB#v=onepage&q=fractal%20unstable%20manifold&f=false

For example for some parametrization of the Henon map the unstable manifold is similar (indeed it is contained in it) to the Henon attractor, and the Henon attractor is a fractal.

Any idea about this?

Answer 1

Invariant "manifolds" are not generally submanifolds in the strict sense of differential topology.

For instance, in a 2-dimensional dynamical system with a fixed point of index 1, the invariant unstable 1-manifold can be described in an appropriate local coordinate chart $U \approx I \times J$ as $I \times C$ for some countable subset $C \subset J$, and this set $C$ need not be discrete as required for a submanifold.

The closure of the invariant manifold will typically be a "lamination", meaning that in an appropriate coordinate chart $U \approx I \times J$ it has the form $I \times \tau$ where $\tau \subset J$ is a closed subset; in the context of the previous paragraph, $\tau$ is the closure of $C$. The "leaves" of this laminations are subsets of the form $I \times t$, $t \in \tau$. This subset $\tau$ could possibly be of fractional Hausdorff dimension, and that is what leads to the possibility that attractors can be fractal.

So, for instance, in your statement the Henon attractor equals the closure of the unstable manifold.