= Solution
For $u>0$, the second-iterate equation $H^2(u)=u$ is equivalent to $u+H(u)=2\log r$. Write $L=\log r$ and let paired points be $L-d,L+d$. The condition $H(L-d)=L+d$ becomes $\tanh(d/2)=d/L$. At $r=9$, $L=2\log3>2$, so <strict concavity> of the positive <hyperbolic tangent> gives one nonzero positive $d$, as well as $d=0$. Here it is exactly $d=\log3$.
Thus the four nonnegative <fixed points> of $H^2$ are
$$
\boxed{0,\quad\log3,\quad2\log3,\quad3\log3.}
$$
Zero has <periodic-orbit multiplier> $81$ and the central point has <periodic-orbit multiplier> $(1-2\log3)^2>1$, so both are unstable. The outer two points are exchanged by $H$, since $H(\log3)=3\log3$ and $H(3\log3)=\log3$. Their common second-iterate <periodic-orbit multiplier> is
$$
M=H'(\log3)H'(3\log3)=(1-\log3)(1-3\log3)\simeq0.226398.
$$
Its magnitude is below one, so both outer <fixed points> of $H^2$ are stable. This verifies the crossings and stability in the supplied graph rather than reading stability from crossing direction alone.
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