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Universidade Federal do Rio Grande do Sul（南里奥格兰德联邦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明在强聚束条件下，周期测度李雅普诺夫谱的一致渐近间隙蕴含控制分裂，将周期间隙判据推广至任意维 Hölder 余环。
AI 中文摘要
设 $A$ 是定义在传递的双边有限型子移位上的 $\alpha$-Hölder 余环，并假设 $A$ 是强聚束的。我们证明，若存在 $k\in\{1,\ldots,d-1\}$ 和 $c>0$，使得对每个周期点 $p$ 都有 $\lambda_k(p)-\lambda_{k+1}(p)\ge c$，则 $A$ 具有指标 $k$ 的控制分裂。因此，在强聚束条件下，周期测度的李雅普诺夫谱中的一致渐近间隙意味着沿所有轨道段存在一致几何分裂。这将对周期间隙判据从局部常数余环推广到任意维数的 Hölder 余环，并将二维纤维聚束结果推广到更强聚束假设下的更高维数。
英文摘要
Let $A$ be an $α$-Hölder cocycle over a transitive two-sided subshift of finite type, and assume that $A$ is strongly bunched. We prove that if there exist $k\in\{1,\ldots,d-1\}$ and $c>0$ such that \[ λ_k(p)-λ_{k+1}(p)\ge c \] for every periodic point $p$, then $A$ admits a dominated splitting of index $k$. Thus, under strong bunching, a uniform asymptotic gap in the Lyapunov spectrum of periodic measures implies a uniform geometric splitting along all orbit segments. This extends the periodic-gap criterion of Kassel--Potrie from locally constant cocycles to Hölder cocycles in arbitrary dimension, and extends the two-dimensional fiber-bunched result of Velozo to higher dimensions under the stronger bunching assumption.
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