For $|\alpha|=1$, the map $z\mapsto cz$ sends $\widehat C_{m,\alpha}$ into itself if and only if $c^{2m}=1$

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For $|\alpha|=1$, the map $z\mapsto cz$ sends $\widehat C_{m,\alpha}$ into itself if and only if $c^{2m}=1$ · Prove2Me