Let $\pi\colon(X,T)\to (Y,T)$ be a factor map between minimal subshifts. Suppose there exists $\tilde{Y} \subseteq Y$ such that
- $\# \pi^{-1}(y) = 1$ for all $y \in \tilde{Y}$.
- $\tilde{Y}$ is a residual subset of $Y$ i.e. $\overline{Y\setminus \tilde{Y}}$ has empty interior$\tilde{Y} = \bigcap_{n=1}^\infty\tilde{Y}_n$ for some collection $\{\tilde{Y}_1,\tilde{Y}_2,\dots\}$ of dense open subsets of $Y$.
- $\mu(\tilde{Y}) = 1$ for every $T$-invariant probability measure in $Y$.
Is it true that $\pi$ is injective?