A lemniscate as the spectrum of a perturbed shift

The spectrum of the perturbed shift operator $T$, $T\colon f(n)\mapsto f(n+1)+a(n)f(n)$, in $\ell^2(\mathbf Z)$ is considered for $a(n)$ taking a finite set of values. It is proved that if all values of the function $a(n)$ have uniform frequencies on $\mathbf Z$, then the essential part of the spectrum fills a generalized lemniscate.