  We consider families of diffeomorphisms with dominated splittings and
preserving a Borel probability measure, and we study the regularity of the
Lyapunov exponents associated to the invariant bundles with respect to the
parameter. We obtain that the regularity is at least the sum of the
regularities of the two invariant bundles (for regularities in $[0,1]$), and
under suitable conditions we obtain formulas for the derivatives. Similar
results are obtained for families of flows, and for the case when the invariant
measure depends on the map.
  We also obtain several applications. Near the time one map of a geodesic flow
of a surface of negative curvature the metric entropy of the volume is
Lipschitz with respect to the parameter. At the time one map of a geodesic flow
on a manifold of constant negative curvature the topological entropy is
differentiable with respect to the parameter, and we give a formula for the
derivative. Under some regularity conditions, the critical points of the
Lyapunov exponent function are non-flat (the second derivative is nonzero for
some families). Also, again under some regularity conditions, the criticality
of the Lyapunov exponent function implies some rigidity of the map, in the
sense that the volume decomposes as a product along the two complimentary
foliations. In particular for area preserving Anosov diffeomorphisms, the only
critical points are the maps smoothly conjugated to the linear map,
corresponding to the global extrema.
