I am looking for an example of a factor $f\colon (X,T) \to (Y,T)$ between topological dynamical systems, where $(X,T)$ is a minimal subshift and $Y$ a connected topological space such that $(Y,T)$ is not a group rotation. Note that any eigenvalue $\lambda = \exp(2\pi i\alpha)$ of $(X,T)$ gives us a factor $(S^1,+\alpha)$ wich is connected and a group rotation; can there be another class?

Let $\alpha$ be irrational, let $Y$ be the two-dimensional torus equipped with the map $S(u,v)=(u+\alpha,v+u)$. Then the action of $S$ on $Y$ is minimal. Now partition the torus into two pieces, say $A_0=S^1\times[0,\frac 12)$ and $A_1=S^1\times[\frac 12,1)$ and let $j(y)=0$ if $y\in A_0$ and $j(y)=1$ if $y\in A_1$. Let $X$ be the set of bi-infinite sequences with symbols 0 and 1 such that each sub-word occurs as a sequence $j(y),\ldots,j(S^{n-1}y)$ for some $y\in \mathbb T^2$ and some $n$.

Define for $x\in X$, $$ \pi(x)=\bigcap_{n\in\mathbb Z}\overline{S^{-n}(A_{x_n})}. $$ This gives a single point of the torus (since the partition is generating). This map intertwines the shift and $S$, and is surjective. The minimality of $S$ acting on $Y$ implies that of the shift on $X$.