# Automorphisms of strictly ergodic shift spaces

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?

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