= Solution
The <Ricker population map two-cycle> consists of the two stable points of the second iterate, exchanged by the original map. Thus generic positive <orbit (dynamical system)>[orbits] approach alternating values
$$
\boxed{\log3\ \longleftrightarrow\ 3\log3.}
$$
There are exceptions: zero stays extinct, while an <orbit (dynamical system)> starting at $\log9$ or at one of its finite preimages eventually stays at that unstable <fixed point>.
To justify the large-time conclusion, put $b=9/e$ and $a=H(b)\simeq1.087>1$. Every first image is at most $b$. A positive <orbit (dynamical system)> below $a$ grows until it enters $[a,b]$; if it never entered, its monotone limit below $a<\log9$ would contradict the fixed-point equation. The interval $[a,b]$ is invariant. On it $H$ is decreasing, so $H^2$ is increasing. Its only <fixed points> there are the three positive points already found; its graph signs make iterations below the unstable middle point converge to the lower stable point, and those above converge to the upper stable point. Interchanging even and odd subsequences under $H$ then gives the stated <two-cycle>. This is regular period-two behaviour, not chaos at this parameter.
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