AI 中文总结
研究紧致平移曲面\(\mathbb{Z}^d\)覆盖上自相似平移流关于马哈拉姆测度的遍历性质，基于扭曲转移算子开发重整化方法，得出马哈拉姆 - 波利科特 - 吕埃勒共振，还给出相关应用成果。
AI 中文摘要
我们研究紧致平移曲面的\(\mathbb{Z}^d\)覆盖上的自相似平移流。主要目标是研究其关于一般马哈拉姆测度的遍历性质。为此，我们基于一族与作用在各向异性分布空间上的重整化伪阿诺索夫映射相关的扭曲转移算子开发了一种重整化方法。我们根据伪阿诺索夫在合适的扭曲上同调群上的作用描述了这些算子的离散谱。还表明对偶算子对应于周边特征值的共振态产生了在平移流下不变的马哈拉姆分布。基于此对应，我们将这些特征值称为马哈拉姆 - 波利科特 - 吕埃勒共振。作为应用，我们推导了马哈拉姆一般点处光滑可观测量的遍历积分的渐近公式，证明了相关弗罗贝尼乌斯余圈的中心极限定理，并计算了马哈拉姆测度的豪斯多夫维数。
英文摘要
We study self-similar translation flows on $\mathbb Z^d$-covers of compact translation surfaces. Our main goal is to investigate their ergodic properties with respect to general Maharam measures. To this end, we develop a renormalization approach based on a family of twisted transfer operators associated with the renormalizing pseudo-Anosov map acting on anisotropic spaces of distributions. We describe the discrete spectrum of these operators in terms of the action of the pseudo-Anosov on suitable twisted cohomology groups. We further show that the resonant states of the dual operator corresponding to peripheral eigenvalues give rise to Maharam distributions which are invariant under the translation flow. Motivated by this correspondence, we refer to these eigenvalues as Maharam-Pollicott-Ruelle resonances. As applications, we derive asymptotic formulas for ergodic integrals of smooth observables at Maharam-generic points, prove a central limit theorem for the associated Frobenius cocycle, and compute the Hausdorff dimension of Maharam measures.
Comments74 pages
