We provide a unified framework to study the nonuniform exponential behavior of a dynamics using the theory of evolution semigroups. We emphasize that nonuniform hyperbolicity is ubiquitous in the context of ergodic theory: for an autonomous differential equation whose flow preserves a finite measure, the variational equations of almost all trajectories with nonzero Lyapunov exponents have a nonuniform exponential behavior. We consider the general case of a nonautonomous dynamics on a Banach space. In particular, we show that to any nonuniformly exponentially bounded evolution family, one can associate a C₀ semigroup (thus an autonomous dynamics) on some appropriate Banach space. This semigroup is then used to characterize completely the nonuniform hyperbolicity of the evolution family. In addition, we establish a corresponding spectral mapping theorem for an evolution semigroup, which allows us to obtain another characterization of the nonuniform hyperbolicity of an evolution family in terms of the spectrum of the evolution semigroup.

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通过演化半群的非均匀指数行为

我们提供了一个统一的框架，以使用演化半群理论研究动力学的非均匀指数行为。我们强调，在遍历理论的背景下，非均匀双曲是普遍存在的：对于一个自治的微分方程，其流动保持有限的度量，几乎所有具有非零Lyapunov指数的轨迹的变分方程都具有非均匀的指数行为。我们考虑Banach空间上非自治动力学的一般情况。特别是，我们表明，对于任何一个非均匀指数有界的进化族，一个人都可以将一个C ₀在适当的Banach空间上的半群（因此具有自主动力学）。然后使用该半群来完全描述进化族的非均匀双曲性。此外，我们为演化半群建立了一个对应的频谱映射定理，这使我们能够根据演化半群的频谱获得演化族的非均匀双曲性的另一个特征。