# Does an asymptotic component with large size in a minimal subshift always exist?

Let $$(X, T)$$ be a minimal subshift, i.e. $$X$$ is a closed $$T$$-invariant subset of $$A^\mathbb{Z}$$, where $$T$$ is the shift. A pair $$x,y\in X$$ is asymptotic if $$d(T^nx, T^ny)$$ goes to zero as $$n\to\infty$$. Always exists such a pair when $$X$$ is infinite: for every $$n\geq1$$ there exists $$x^{(n)}, y^{(n)} \in X$$ such that $$x^{(n)}_0\not= y^{(n)}_0$$ and $$x^{(n)}_{[1,N]}= y^{(n)}_{[1,N]}$$ (if not, for some $$n$$, $$x_{[1,n]} = y_{[1,n]}$$ implies $$x_0=y_0$$, i.e., $$x_{[1,n]}$$ determines $$x_0$$, and this forces $$X$$ to be periodic), and any pair of convergent subsequences of $$x^{(n)}$$ and $$y^{(n)}$$ will do the trick.

My question is: do exist $$k$$-tuples of asymptotic points, for every $$k\geq1$$? More precisely, is it true that for every $$k\geq1$$ there exists $$x_1,\dots,x_k\in X$$ such that $$\lim_{n\to\infty}d(T^nx_i, T^nx_j) = 0\ \forall i,j$$

• No. Consider the Sturmian shift. There asymptotic pairs, but no triples. – Anthony Quas May 16 '19 at 6:23