# Example of connected factor of symbolic system that is not a rotation

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 $$X$$.