  We continue our study of the dynamics of meromorphic mappings with small
topological degree on a compact K\"ahler surface $X$. Under general hypotheses
we are able to construct a canonical invariant measure which is mixing, does
not charge pluripolar sets and admits a natural geometric description.
  Our hypotheses are always satisfied when $X$ has Kodaira dimension zero, or
when the mapping is induced by a polynomial endomorphism of $\mathbf{C}^2$.
They are new even in the birational case. We also exhibit families of mappings
where our assumptions are generically satisfied and show that if
counterexamples exist, the corresponding measure must give mass to a pluripolar
set.
