Isometries with Dense Windings of the Torus in $C(M)$

Let $C(M)$ be the space of all continuous functions on $M\subset\mathbb{C}$. We consider the multiplication operator $T\colon C(M)\to C(M)$ defined by $Tf(z)=zf(z)$ and the torus $O(M)=\{f:M\to\mathbb{C},\, \|f\|=\|\frac{1}{f}\|=1\}$. If $M$ is a Kronecker set, then the $T$-orbits of the points of the torus $\frac12 O(M)$ are dense in $\frac12 O(M)$ and are $\frac12$-dense in the unit ball of $C(M)$.