AI 中文总结
该研究构造了一类闭流形上的保体积同胚，实现了Hölder可观测量的相关性指数衰减，肯定了Dolgopyat和Pesin的相关猜想，还给出了无穷多个无对应微分同胚的此类流形实例。
AI 中文摘要
我们证明，每一个维数$n\ge 4$的闭流形，若存在奇异2-叶状结构（即由闭曲面构成且其商空间为穿孔$(n-2)$-环面的叶状结构），则该流形上存在保体积同胚，其对Hölder可观测量的相关性呈指数衰减。该证明引入了一类称为相对Anosov同胚的系统，这类系统仅在奇异集处不可微。我们应用一个一般性二分法，该二分法将任意保不变叶状结构的同胚的相关性衰减率，用商动力学的衰减率与叶间循环的衰减率的最大值来界定。此二分法本身具有独立意义，作为直接推论可给出乘积系统、斜乘积、具有紧中心叶的部分双曲微分同胚、有限覆盖及由纤维映射扩张的系统的衰减率。随后我们证明，这一结果对Dolgopyat和Pesin提出的关于在一大类流形上实现相关性指数衰减的猜想给出了肯定答案，此前这类流形中尚无已知系统具有相关性指数衰减。特别地，我们构造了无穷多个两两不同胚的闭4-流形，这些流形上存在保体积同胚，其相关性呈指数衰减，但这类流形不支持Anosov微分同胚或强部分双曲微分同胚。
英文摘要
We prove that every closed manifold of dimension $n\ge 4$ which admits a singular $2$-foliation (a foliation by closed surfaces whose quotient is a punctured $(n-2)$-torus) supports a volume-preserving homeomorphism with exponential decay of correlations for Hölder observables. The proof introduces a class of systems called relatively Anosov homeomorphisms, which fails differentiability only at the singular set. We apply a general dichotomy that bounds the decay of correlations of any homeomorphism preserving an invariant foliation by the maximum of the decay rates of the quotient dynamics and of the leafwise cycles. This dichotomy is of independent interest and yields, as immediate consequences, decay rates for product systems, skew products, partially hyperbolic diffeomorphisms with compact center leaves, finite covers, and extensions by expanding fibre maps. We then prove this gives a positive answer to a conjecture of Dolgopyat and Pesin regarding the realization problem for exponential decay of correlations over a large class of manifolds for which there were no known systems with exponential decay of correlations. In particular, we construct infinitely many pairwise non-homeomorphic closed $4$-manifolds which admit a volume-preserving homeomorphism with exponential decay of correlations, albeit not supporting either Anosov diffeomorphisms or strong partially hyperbolic diffeomorphisms.
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