Bounds of the spectral radius of the induced map on cohomology

TL;DR

This paper explores how the spectral radius of the induced map on cohomology relates to the existence of invariant measures with positive or negative Lyapunov exponents in $C^{1}$-diffeomorphisms on compact manifolds, supporting aspects of Shub's entropy conjecture.

Contribution

It establishes conditions under which the spectral radius of the induced cohomology map guarantees positive or negative Lyapunov exponents for invariant measures, extending previous results.

Findings

Spectral radius > 1 implies existence of invariant measure with positive Lyapunov exponent.

Volume-preserving case shows existence of measures with both positive and negative exponents.

Under integrability conditions, positive volume sets with non-zero Lyapunov exponents are guaranteed.

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.