12
$\begingroup$

Let $(\{1,...,2^{n}\},*_{n})$ denote the $n$-th Laver table.

Let $$C_{n}=\{(\frac{x}{2^{n}},\frac{x*_{n}y}{2^{n}})|x,y\in\{1,2,3,...,2^{n}\}\}$$ for all $n\in\mathbb{N}$.

Then since $C_{n}$ is a finite subset of $[0,1]^{2}$, one can talk about the convergence of $C_{n}$ in the Hausdorff metric (i.e. the convergence in the hyperspace topology or the Vietoris topology).

Does $(C_{n})_{n\in\mathbb{N}}$ converge in the Hausdorff metric to $S\cup L$ where $S$ is the Sierpinski triangle in the upper-left half of $[0,1]\times[0,1]$ and $L=[0,1]\times\{1\}$?

Computer calculations seem to suggest that $C_{n}\rightarrow S\cup L$.

$\endgroup$
  • $\begingroup$ On this page boolesrings.org/jvanname/…, we can reorder the elements of $A_{n}$ in a canonical way so that the compact set $B_{n}$ corresponding to $C_{n}$ (composed of squares instead of points) are strictly decreasing and therefore converge to a compact set $B$. However, the compact $B$ is a proper subset of a Sierpinski-like fractal. $\endgroup$ – Joseph Van Name Dec 1 '16 at 15:54

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.