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

Abstract

Let C(M) be the space of all continuous functions on M⊂ ℂ. 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).