  Let $f:M\rightarrow M$ be a $C^1$ diffeomorphism with a dominated splitting
on a compact Riemanian manifold $M$ without boundary.
  We state and prove several sufficient conditions for the topological entropy
of $f$ to be positive. The conditions deal with the dynamical behaviour of the
(non-necessarily invariant) Lebesgue measure. In particular, if the Lebesgue
measure is $\delta$-recurrent then the entropy of $f$ is positive. We give
counterexamples showing that these sufficient conditions are not necessary.
Finally, in the case of partially hyperbolic diffeomorphisms, we give a
positive lower bound for the entropy relating it with the dimension of the
unstable and stable sub-bundles.
