  The notion of flat $\lambda$-connections as the interpolation of usual flat
connections and Higgs fields was suggested by Deligne and further studied by
Simpson. Mochizuki established the Kobayashi--Hitchin-type theorem for
$\lambda$-flat bundles ($\lambda\neq 0$), which is called the Mochizuki
correspondence. In this paper, on the one hand, we generalize Mochizuki's
result to the case when the base being a compact balanced manifold, more
precisely, we prove the existence of harmonic metrics on stable $\lambda$-flat
bundles ($\lambda\neq 0$). On the other hand, we study two applications of the
Simpson--Mochizuki correspondence to moduli spaces. More concretely, we show
this correspondence provides a homeomorphism between the moduli space of
(semi)stable $\lambda$-flat bundles over a complex projective manifold and the
Dolbeault moduli space, and also provides dynamical systems with two parameters
on the latter moduli space. We investigate such dynamical systems, in
particular, we calculate the first variation, the fixed points and discuss the
asymptotic behaviour.
