  Let $\{F_n, n\ge 8\}$ be a family of diffeomorphisms on real rational
surfaces that are birationally equivalent to birational maps on
$\mathbf{P}^2(\mathbb{R})$. In this article, we investigate the mapping classes
of the diffeomorphisms $F_n, n\ge 8$. These diffeomorphisms are reducible with
unique invariant irreducible curves, and we determine the mapping classes of
their restrictions, $\hat F_n, n \ge 8$, on the cut surfaces, showing that they
are pseudo-Anosov and do not arise from Penner's construction. For $n=8$,
Lehmer's number is realized as the stretch factor of $\hat F_8$, a
pseudo-Anosov map on a once-punctured genus $5$ orientable surface. The
diffeomorphism $\hat F_8$ is a new geometric realization of a Lehmer's number.
