The recurrence protocol's singlet fraction map is f_ s ' = f_ s ^2+ 19(1-f_ s )^2 f_…

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Problem

The recurrence protocol's singlet fraction map is fs′=fs2+19(1−fs)2fs2+23fs(1−fs)+59(1−fs)2.f_{\rm s}' = \frac{f_{\rm s}^2+\tfrac19(1-f_{\rm s})^2}{f_{\rm s}^2+\tfrac23f_{\rm s}(1-f_{\rm s})+\tfrac59(1-f_{\rm s})^2}. ((a)) Show that fs=14f_{\rm s}=\tfrac14, 12\tfrac12 and 11 are each a fixed point of this map, meaning fs′=fsf_{\rm s}'=f_{\rm s} there. ((b)) Compute dfs′/dfsdf_{\rm s}'/df_{\rm s} at each of the three fixed points, and use the sizes found to decide which fixed points are stable under repeated rounds and which is unstable.

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