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Since the product of the eigenvalues of a fixed point equals $b<1$, the eigenvalues must be real when one of them "crosses" from $\left\vert\lambda\right\vert<1$ to $\left\vert\lambda\right\vert>1$.

So we wish to examine what happens when $\left\vert \lambda \right\vert = 1$. There are two cases since $\lambda $ is real: $\lambda =1$ and $\lambda=-1$.

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