  All the periodic points of a certain algebraic function related to the
Rogers-Ramanujan continued fraction $r(\tau)$ are determined. They turn out to
be $0, \frac{-1 \pm \sqrt{5}}{2}$, and the conjugates over $\mathbb{Q}$ of the
values $r(w_d/5)$, where $w_d$ is one of a specific set of algebraic integers,
divisible by the square of a prime divisor of 5, in the field
$K_d=\mathbb{Q}(\sqrt{-d})$, as $-d$ ranges over all negative quadratic
discriminants for which $\left(\frac{-d}{5}\right) = +1$. This yields new
insights on class numbers of orders in the fields $K_d$. Conjecture 1 of Part I
is proved for the prime $p=5$, showing that the ring class fields over fields
of type $K_d$ whose conductors are relatively prime to $5$ coincide with the
fields generated over $\mathbb{Q}$ by the periodic points (excluding -1) of a
fixed $5$-adic algebraic function.
