上同调诱导映射的谱半径的界
Bounds of the spectral radius of the induced map on cohomology
摘要：
本文研究了紧致流形上$C^{1}-$代数同态的上同调诱导映射与李雅普诺夫指数之间的关系。证明了如果上同调映射的谱半径严格大于1，则其上同调映射的不变遍历测度至少有一个正的李雅普诺夫指数.此外，如果复同态也保持一个连续的体积形式，那么它有一个不变的遍历测度，至少有一个正的和一个负的李雅普诺夫指数，与舒布的熵猜想一致。我们还考虑了保持一个相当于体积的测度的双同态。在这种情况下，我们证明了如果Lyapunov度量满足可积性条件，且上同调诱导映射的谱半径严格大于1，则在正体积集上，该映射的李雅普诺夫指数不为零.
英文摘要：
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.
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