It is well known that Toeplitz subshifts are minimal almost 1-to-1 extensions of an odometer, and that some of these subshifts have positive entropy. Thus, even if a system is an almost 1-to-1 extension of an equicontinuous system, it can be very complicated. What other examples of this behavior are known?
More precisely, I ask the following: is there a concrete example of a minimal system having positive entropy which is an almost 1-to-1 extension of an irrational rotation $((\mathbb{R}/\mathbb{Z})^n,+\vec{\alpha})$?
