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

A characterization of those points in the unit disc which belong to the spectrum of a composition operator $C_φ$, defined by a rotation $φ(z)=rz$ with $|r|=1$, on the space $H_0(\mathbb{D})$ of all analytic functions on the unit disc which vanish at $0$, is given. Examples show that the point $1$ may or may not belong to the spectrum of $C_φ$, and this is related to Diophantine approximation. Our results complement recent work by Arendt, Celariès and Chalendar.