Let $(X,T)$ be a minimal subshift. Can it happen that an endomorphism $\varphi\colon (X,T) \to (X,T)$ is almost 1-to-1 but not 1-to-1? Can it happen that a factor $\pi\colon (X,T) \to (Y,T)$ between minimal subshifts is almost 1-to-1 but not 1-to-1? I know that, for example, Toeplitz subshifts are almost 1-to-1 extensions of a Cantor system (some odometer), but in the subshift case I couldn't find anything on internet.


PD. Here, a factor $\pi\colon(X,T)\to(Y,T)$ is almost 1-to-1 if $\exists y \in Y$ such that $\#\pi^{-1}(y) = 1$. From this, one can prove that $\#\pi^{-1}(y) = 1$ for all $y$ in a residual set.



Almost one-to-one endomorphism of minimal subshift?
The answer is yes. In [this paper](https://link.springer.com/article/10.1007/BF02774039) Downarowicz proves the following theorem



> 
> There exists a regular Toeplitz sequence $\omega$ such that the induced
>  Toeplitz flow $(\bar O(\omega), S)$ is noncoalescent, more precisely, it admits an endomorphism $\pi : \bar{O}(\omega) \to \bar{O}(\omega)$ of the first type.
> 
> 
> 


Here, $\bar O(\omega)$ is the orbit closure (so a Toeplitz subshift since $\omega$ is Toeplitz), $S$ the shift map, **noncoalescent** means not injective, and **first type** means every Toeplitz point has a unique preimage, in particular some point does.


