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A characterization of those points in the unit disc which belong to thespectrum of a composition operator $C_{\varphi}$, defined by a rotation$\varphi(z)=rz$ with $|r|=1$, on the space $H_0(\mathbb{D})$ of all analyticfunctions on the unit disc which vanish at $0$, is given. Examples show thatthe point $1$ may or may not belong to the spectrum of $C_{\varphi}$, and thisis related to Diophantine approximation. Our results complement recent work byArendt, Celari\`es and Chalendar.

The spectrum of composition operators induced by a rotation in the space of all analytic functions on the disc