AI 中文总结
本文针对具有连续不变和乐族的上循环，证明李雅普诺夫指数仍可通过周期轨道近似，该几何方法的证明比现有文献更简洁直接。
AI 中文摘要
经典结论表明，对于双曲系统上的线性上循环，若其满足赫尔德连续条件，则遍历测度的李雅普诺夫指数可通过周期轨道的李雅普诺夫指数近似。Bochi近期的反例显示，若将赫尔德假设放宽为仅连续，该近似性质一般不成立。本文引入一种几何条件，可成功替代该分析正则性假设。更确切地说，我们证明：若上循环（即使是不连续的）具有连续的不变和乐族，李雅普诺夫指数的周期近似仍然成立。我们的几何方法所得证明比文献中现有论证更简洁直接，即便应用于纤维束上循环等经典情形（此前已有相关结果）亦是如此。
英文摘要
Classical results establish that the Lyapunov exponents of an ergodic measure for linear cocycles over hyperbolic systems can be approximated by the Lyapunov exponents of periodic orbits, provided the cocycle is Hölder continuous. A recent counterexample by Bochi demonstrates that this approximation property fails in general if the Hölder assumption is relaxed to mere continuity. In this paper, we introduce a geometric condition that successfully substitutes this analytical regularity hypothesis. More precisely, we prove that if a cocycle - even a discontinuous one - admits a continuous family of invariant holonomies, the periodic approximation of Lyapunov exponents still holds. Our geometric approach yields a proof that is substantially simpler and more direct than existing arguments in the literature, even when applied to classical settings such as fiber-bunched cocycles for which previous results were already available.
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