Obstacles to Topological Factoring of Toeplitz shifts

For every Toeplitz sequence $x$ with period structure $(q_i{)}_{i\ge 1}$, one can identify a period structure $\mathbf{p}=(p_i{)}_{i\ge 0}$ which leads to a Bratteli-Vershik realization of the associated Toeplitz shift; we refer to this period structure as {\it constructive}. Let $(X,\sigma ,x)$ and $(Y,\sigma ,y)$ be Toeplitz shifts where $x\in X$ and $y\in Y$ are Toeplitz sequences with constructive period structures $(p^n{)}_{n\ge 1}$ and $(q^n{)}_{n\ge 1}$, respectively. Using the Bratteli-Vershik realization of factor maps between Toeplitz shifts, we prove that if there exists a topological factoring $\pi :(X,\sigma )\to (Y,\sigma )$ with $\pi (x)=y$, then $q\mid p$. In particular, if $\pi$ is conjugacy, then $p=q$. We also prove that Toeplitz sequences are mapped to Toeplitz sequences through topological factorings.