  We consider partially hyperbolic attractors for non-singular endomorphisms
admitting an invariant stable bundle and a positively invariant cone field with
non-uniform cone expansion at a positive Lebesgue measure set of points. We
prove volume lemmas for both Lebesgue measure on the topological basin of the
attractor and the SRB measure supported on the attractor.As a consequence under
a mild assumption we prove exponential large deviation bounds for the
convergence of Birkhoff averages associated to continuous observables with
respect to the SRB measure.
