Let $\pi\colon(X,T)\to (Y,T)$ be a factor map between minimal subshifts. Suppose there exists $\tilde{Y} \subseteq Y$ such that


1. $\# \pi^{-1}(y) = 1$ for all $y \in \tilde{Y}$.
2. $\tilde{Y}$ is a residual subset of $Y$ *i.e.* $\tilde{Y} = \bigcap\_{n=1}^\infty\tilde{Y}\_n$ for some collection $\{\tilde{Y}\_1,\tilde{Y}\_2,\dots\}$ of dense open subsets of $Y$.
3. $\mu(\tilde{Y}) = 1$ for every $T$-invariant probability measure in $Y$.


Is it true that $\pi$ is injective?



Does this strong form of being almost 1-to-1 imply injectivity?
No. Consider an irrational rotation $R$ of the circle (which I identify with [0,1)) by an angle $\alpha$. Let $\alpha<\beta<1$ be a point not lying in the orbit of 0 under $R$. Set $A\_1=[0,\alpha)$, $A\_2=[\alpha,\beta)$, $A\_3=[\beta,1)$ also $B\_1=A\_1$ and $B\_2=A\_2\cup A\_3$.
Consider the partitions $P=\{A\_1,A\_2,A\_3\}$ and $Q=\{B\_1,B\_2\}$.
For $x\in S^1$, let $\epsilon\_1(x)\in \{1,2,3\}^{\mathbb Z}$ be such that $R^n(x)\in A\_{\epsilon\_1(x)\_n}$ and similarly let $\epsilon\_2(x)\in \{1,2\}^{\mathbb Z}$ be such that 
$R^n(x)\in B\_{\epsilon\_2(x)\_n}$. 


Let $X$ be the (minimal) subshift of $\{1,2,3\}^{\mathbb Z}$ consisting of the orbit closure of $\{\epsilon\_1(x)\colon x\in [0,1)\}$ and $Y$ be the orbit closure of $\{\epsilon\_2(x)\colon x\in [0,1)\}$. There is an obvious factor map from $X$ to $Y$ (replacing 3 symbols by 2's). This factor map is 1-1 off a countable set (corresponding to orbits passing through $\beta$). 


