**abstract:**
Given a product $G$ of noncompact simple Lie groups, a closed subgroup
$H \subset G$ and an irreducible lattice
$\Gamma \subset G$, Moore's Ergodicity Theorem implies that the action
of $\Gamma$ on the homogeneous space $G*H$ is ergodic if and only if
$H$ is noncompact. We apply this ergodic phenomenon to rigidity
problems on the (finite-volume) quotient $X := \Gamma \backslash \Omega$
of a bounded symmetric domain $\Omega$ of rank $\ge 2$ by a torsion-free irreducible
lattice $\Gamma \subset \text{Aut}(\Omega)$,
thereby showing that any nontrivial holomorphic map $f: X \to N$
into a quotient $N$ of {\it any} bounded domain $D$ must lift to a holomorphic
embedding $F: \Omega \to D$. The proof makes use of Moore's
Ergodicity Theorem in conjunction with bounded
holomorphic functions which are extremal with respect to Carathéodory-like
complex Finsler metrics. As an application of the proof we show that any
nontrivial proper holomorphic mapping $f: X \to Z$ of $X$ onto a
complex space $Z$ must be an unramified covering unless the fundamental
group $\pi _{1}(Z)$ of $Z$ is finite. Regarding the embedding result, on top
of the proof using intrinsic complex Finsler metrics there is an
alternative proof which makes use
of Moore's Ergodicity Theorem in conjunction with non-tangential boundary
values of bounded holomorphic functions on $\Omega$. This
alternative proof yields stronger results by producing at the same time
a left inverse $R: D \to \Bbb C^{n$} of the embedding $F$ (i.e., $R \circ F = \text{id}_{{\Omega}$)
}
as a bounded holomorphic map. *

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