Local stability derivatives at the two fixed points

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Differentiating the average recurrence gives (5−z⁻²)/7. Its values at one and minus one half are 4/7 and 1/7, both with magnitude below one, so both fixed points are locally attracting for the scalar iteration. Local attraction does not prove global convergence across the singularity at zero.

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Original context

zfixed,1=1,  zfixed,2=−12.

The derivative is

F′z=5−z−27,  F′1=47,  F′−1/2=17.

Both fixed points are locally attracting. This statement is local for the negative fixed point; the proof below establishes global convergence only for positive starts.

6.3 Proposition 3: every positive averaged trajectory converges to one

Two factorizations establish the result:

F′\left(z\right)=\frac{5-{z}^{-2}}{7},  F′\left(1\right)=\frac{4}{7},  F′\left(-1/2\right)=\frac{1}{7}.