Let $X$ be a strictly ergodic shift space, and $\omega_1$, $\omega_2$ be two different points in $X$. Is there an automorphism $\Psi$ of $X$ such that $\Psi(\omega_1)=\omega_2$? By an automorphism I mean a shift-commuting homeomorphism of $X$. The answer for a general minimal shift space is, I guess, negative as there are minimal shift spaces with two non-isomorphic ergodic measures. But what if $X$ posses only one ergodic measure?
Strict ergodicity does not seem to change much, i.e. the answer is still negative, see the following theorem (W. Bułatek, J. Kwiatkowski, Strictly ergodic Toeplitz flows with positive entropies and trivial centralizers)
The exist a strictly ergodic Toeplitz flows with trivial centralizer.
By a Toeplitz flow we mean a closure of the orbit of a Toeplitz sequence, together with the shift map. Since any block appears on a Toeplitz sequence with some period, it follows that the unique invariant measure has full support, hence minimality.