  We establish an averaging principle for a family of
solutions$(X^{\varepsilon}, Y^{\varepsilon})$ $ :=$
$(X^{1,\,\varepsilon},\,X^{2,\,\varepsilon},\, Y^{\varepsilon})$ of a system of
SDE-BSDEwith a null recurrent fast component $X^{1,\,\varepsilon}$. Incontrast
to the classical periodic case, we can not rely on aninvariant probability and
the slow forward component$X^{2,\,\varepsilon}$ cannot be approximated by a
diffusion process.On the other hand, we assume that the coefficients admit a
limit in a\`{C}esaro sense. In such a case, the limit coefficients may
havediscontinuity. We show that we can approximate the
triplet$(X^{1,\,\varepsilon},\, X^{2,\,\varepsilon},\, Y^{\varepsilon})$ bya
system of SDE-BSDE $(X^1, X^2, Y)$ where $X := (X^1, X^2)$ is aMarkov diffusion
which is the unique (in law) weak solution of theaveraged forward component and
$Y$ is the unique solution to the averaged backward component. This is done
with a backward component whosegenerator depends on the variable $z$.
Asapplication, we establish an homogenization result for semilinearPDEs when
the coefficients can be neither periodic nor ergodic. Weshow that the averaged
BDSE is related to the averaged PDE via aprobabilistic representation of the
(unique) Sobolev $W\_{d+1,\text{loc}}^{1,2}(\R\_+\times\R^d)$--solution of the
limitPDEs. Our approach combines PDE methods and probabilistic argumentswhich
are based on stability property and weak convergence of BSDEsin the S-topology.
