Bounds of the spectral radius of the induced map on cohomology

Abstract

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.