A SUMMARY OF PUGH-SHUB CONJECTURE WITH 2-DIMENSION CENTRAL BUNDLE

In the last 30 years, there is a famous conjecture in the field of partially hyperbolic dynamics: Pugh-Shub conjecture. This conjecture claims that ergodicity abounds in partially hyperbolicity and has not been proved so far. This paper summarizes the proof of Pugh-Shub conjecture when the central dimension is 2. The proof cleverly used the topological tool blender.
The proof of Pugh shub conjecture is not easy. However, ergodicity has been proved to be dense in Ansov, and partially hyperbolic diffeomorphism has a extra central subbundle that neither expands nor contracts than Anosov diffeomorphism, which is called central bundle. Therefore, in recent twenty or thirty years, mathematicians have tried
to prove the strict version of Pugh-Shub conjecture by adding restrictions on the central bundle.
The idea of proof summarized in this paper has also been used by other mathematics workers, so this is indeed an innovative and enlightening proof.

在最近30 年间，在部分双曲领域有一个著名的猜想——Pugh-Shub 猜想。这个
猜想宣称遍历性在部分双曲中大量存在，并至今犹未证明。本文概述了Pugh-Shub猜想在中心维为2 时的证明。该证明巧妙地使用了混合子这个拓扑工具。
Pugh-Shub 猜想的证明并不容易。但遍历性已被证明在双曲中是稠密的，而部
分双曲比双曲多了一个既不扩张也不收缩的中心子丛，被称为中心丛。故近二三
十年间，数学家们都试图通过在中心丛上添加限制的方式去证明Pugh-Shub 猜想
的严格版本。
本文总结的证明思路在之后也被其他数学工作者所使用，故这的确是一个有
创新性、启发性的证明。

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