适应Anosov系统的巴拿赫空间
Banach spaces adapted to Anosov systems
摘要：
本文研究了一类高光滑函数上与Anosov映射相关的Ruelle-Perron-Frobenius算子的谱性质。为此，我们构造各向异性的巴拿赫空间的分布上的转移算子有一个小的本质谱。在C^ \infty的情况下，基本谱半径是任意小的，这产生了具有任意精度的相关性的描述。此外，我们得到了确定性和随机扰动的谱稳定性结果。特别是，我们得到的频谱数据的可微性结果（这意味着可微的SRB措施，CLT的方差，平滑可观察的衰减率等）。
英文摘要：
We study the spectral properties of the Ruelle-Perron-Frobenius operator associated to an Anosov map on classes of functions with high smoothness. To this end we construct anisotropic Banach spaces of distributions on which the transfer operator has a small essential spectrum. In the C^\infty case, the essential spectral radius is arbitrarily small, which yields a description of the correlations with arbitrary precision. Moreover, we obtain sharp spectral stability results for deterministic and random perturbations. In particular, we obtain differentiability results for spectral data (which imply differentiability of the SRB measure, the variance for the CLT, the rates of decay for smooth observable, etc.).
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