Why don't fractals have more differentiable symmetries?

Solution 1:

I do not have a detailed answer, but there is a substantial literature on smooth rigidity of Cantor sets (and other dynamically defined fractals), starting with

D. Sullivan, Differentiable structure on fractal-like sets determined by intrinsic scaling functions on dual Cantor sets. Nonlinear evolution and chaotic phenomena (Noto, 1987), 101–110, NATO Adv. Sci. Inst. Ser. B Phys., 176, Plenum, New York, 1988.

and

D. Sullivan, Differentiable structures on fractal-like sets, determined by intrinsic scaling functions on dual Cantor sets. The mathematical heritage of Hermann Weyl (Durham, NC, 1987), 15–23, Proc. Sympos. Pure Math., 48, Amer. Math. Soc., Providence, RI, 1988.

See for instance:

R. Bamón, C. Moreira, S. Plaza, J. Vera,
Differentiable structures of central Cantor sets.
Ergodic Theory Dynam. Systems 17 (1997), no. 5, 1027–1042.

These papers mainly deal with smooth maps between different Cantor sets, but you should be able to use their results in the setting of a single Cantor set $C$ where you have the extra requirement that $f: C\to C$ sends $x$ to $y$, where $x, y$ are given points.

If you go to www.ams.org/mathscinet and look for papers which refer to the two papers by Sullivan's listed above, you will find many more references.

Solution 2:

I've had better luck searching the literature for results related to the Hilbert-Smith conjecture. The frontier between possible and impossible seems to be somewhere between Lipschitz ambient homogeneity and $C^1$ ambient homogeneity.

On the positive side, from Repovs and Scepin (1997), "A proof of the Hilbert-Smith conjecture for actions by Lipschitz maps":

Malesic proved in 1994 that the standard Cantor set in R2 is Lipschitz ambient homogeneous. He also constructed Antoine’s necklace in R3 which is also Lipschitz ambiently homogeneous [18].

The reference [18] is to J. Malesic: Toroidal decompositions of the 3-dimensional sphere. Ph.D. Thesis, University of Ljubljana, 1995. I can't seem to find this reference, but I can guess with high confidence what they mean.

On the negative side, from Repovs, Skopenkov and Scepin (1996), "$C^1$-homogeneous compacta in $\mathbb R^n$ are $C^1$-submanifolds of $\mathbb R^n$":

We begin by recalling ... that a subset $K \subset \mathbb R^n$ is said to be $C^1$- homogeneous if for every pair of points $x, y \in K$ there exist neighborhoods $O_x, O_y \subset \mathbb R^n$ of $x$ and $y$, respectively, and a $C^1$-diffeomorphism $h: (O_x, O_x \cap K, x) \to (O_y, O_y \cap K, y)$, i.e. $h$ and $h^{-1}$ have continuous first derivatives...

Theorem 1.1. Let $K$ be a locally compact (possibly nonclosed) subset of $\mathbb R^n$. Then $K$ is $C^1$-homogeneous if and only if $K$ is a $C^1$-submanifold of $\mathbb R^n$.

I haven't digested the latter proof, but I think it leaves open the question of whether a Cantor set can be "merely differentiably homogeneous". I can now prove that this is impossible in $\mathbb R^n$ for $n=1$ and $n=2$ by elementary methods, although I don't have a proof for $n\geq 3$.

Further references or clarifications of these results would still be welcome!

Edit: I now have a proof for all $n$, to appear in Topology and its Applications as https://doi.org/10.1016/j.topol.2019.06.046 "Cantor sets are not tangent homogeneous"!