Dispersion properties of ergodic translations

If the rotation angle α is irrational, then ζ is generating for T (see [4]) and the partition ζn = ζ ∨Tζ ∨···∨Tn−1ζ is made out of 2n arcs. This can be easily realized by induction: when passing from ζn−1 to ζn one has to add to the endpoints of the arcs belonging to ζn−1 the two new points Tn(0) and Tn(1/2) (for rational α, say α = p/q, the partition ζn has precisely 2q arcs for all n≥ q). Thus, the rotation is metrically isomorphic to the subshift given by the closure of π([0,1)) where the coding map π : [0,1]→ {−1,1}Z can be defined by π(x)n = θ(Tn(x)) and θ : X → {−1,1} is the function in L2(X ,μ) given by