We show that a \(\operatorname{GL}(d,\mathbb{R})\) cocycle over a hyperbolic system with constant periodic data has a dominated splitting whenever the periodic data indicates it should. This implies global periodic data rigidity of generic Anosov automorphisms of \(\mathbb{T}^{d}\). Further, our approach also works when the periodic data is narrow, that is, sufficiently close to constant. We can show global periodic data rigidity for certain non-linear Anosov diffeomorphisms in a neighborhood of an irreducible Anosov automorphism with simple spectrum.

中文翻译：

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恒定周期数据的主导分裂和阿诺索夫自同构的全局刚性

我们证明，只要周期性数据表明它应该出现，在具有恒定周期数据的双曲系统上的\(\operatorname{GL}(d,\mathbb{R})\)余循环就会出现主导分裂。这意味着\(\mathbb{T}^{d}\)的通用阿诺索夫自同构的全局周期性数据刚性。此外，当周期性数据很窄（即足够接近常数）时，我们的方法也适用。我们可以在具有简单谱的不可约阿诺索夫自同构的邻域中显示某些非线性阿诺索夫微分同胚的全局周期数据刚性。