Simpson–Mochizuki correspondence for λ-flat bundles

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