Isometries with dense windings of the torus in C(M)

Let C(M) be the space of all continuous functions on M? ?. We consider the multiplication operator T: C(M) → C(M) defined by Tf(z) = zf(z) and the torus

$$O(M) = left{ {f:M to mathbb{C} ntrianglelefteq left| f right| = left| {frac{1}{f}} right| = 1} right}$$

. If M is a Kronecker set, then the T-orbits of the points of the torus ½O(M) are dense in ½O(M) and are ½-dense in the unit ball of C(M).