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

Bounds of the spectral radius of the induced map on cohomology