Examples of smooth fractals

Question

A classic example of a fractal curve is the Koch Snowflake. This is a topological manifold (as opposed to many other fractals which are not), but it also clearly not smooth.

Question: Are there any curve-type fractals that are actually smooth? Or does the infinite self-similarity eventually pose an insurmountable barrier to smoothness?

Technically speaking, $\mathbb{R}$ is a smooth fractal too, so for the above question, I'd only introduce the caveat that the curve be 'interesting' as a fractal (or at least non-trivial).

Intuitively, I see no reason for such objects to not exist, but this is far from any area of math I'm familiar with.

Answer 1

I think, it depends on the precise notion of self-similarity and smoothness in your question. In my answer I consider planar curves which are self-similar with respect to affine transformations. De Rham curves provide examples of self-similar curves in this sense. Parabola is self-similar and smooth. More interestingly, there are other de Rham curves which are merely $C^1$-smooth, see the discussion in:

V.Protasov, On the regularity of de Rham curves, Izvestia Math., 2007

freely available here.

My guess is that the only $C^2$-smooth affine self-similar planar curves are algebraic of degree $\le 2$.

Update. It took me awhile to find proper references, but here it goes. First of all, by a self-similar subset of $E^n$ I will mean the limit set $\Lambda$ of an "iterated functional system" of a collection of contracting maps $S_1,...,S_k$ of $E^n$, which belong to some group $G$ of transformations of the Euclidean space or the extended Euclidean space. The most commonly considered groups are:

1. The group $Sim(E^n)$ os Euclidean similitudes (compositions of Euclidean rigid motions and dilations).

2. The group $Aff(E^n)$ of affine transformations.

3. The group $Mob(S^n)$ of Moebius transformations.

I will restrict to the case $n=2$, just for simplicity, much of what I will say holds in higher dimensions as well.

1. In the case $G=Sim(E^2)$, if the self-similar set $\Lambda$ is a differentiable curve (or, even weaker is a curve of Hausdorff dimension 1) then $\Lambda$ is a subset of a straight line. See

V. Mayer, M. Urbański, Finer geometric rigidity of limit sets of conformal IFS. Proc. Amer. Math. Soc. 131 (2003), no. 12, 3695–3702.

2. In the case $G=Aff(E^2)$, the self-similar set can be $C^1$-smooth (see above), but if it is $C^2$ then it has to be contained in a straight line or a parabola, see

C. Bandt, A. Kravchenko, Differentiability of fractal curves. Nonlinearity 24 (2011), no. 10, 2717–2728.

3. In the case $G=Mob(S^2)$ if a self-similar set is a differentiable curve (or, even weaker is a curve of Hausdorff dimension 1) then $\Lambda$ is a subset of a straight line or of a round circle. See the reference in 1.