  In this paper we study the relationship between Lyapunov exponents and the
induced map on cohomology for $C^{1}-$diffeomorphisms on compact manifolds. We
show that if the induced map on cohomology has spectral radius strictly larger
than 1, then the diffeomorphism has an invariant ergodic measure with at least
one positive Lyapunov exponent. Furthermore, if the diffeomorphism also
preserves a continuous volume form then it has an invariant ergodic measure
with at least one positive and one negative Lyapunov exponent, in agreement
with Shub's entropy conjecture. We also consider diffeomorphisms preserving a
measure equivalent to volume. In this case we show that if the Lyapunov metric
satisfies an integrability condition and the induced map on cohomology has
spectral radius strictly larger than 1, then the diffeomorphism has non-zero
Lyapunov exponents on a set of positive volume.
