再论Pesin理论中的$C^{1+α}$假说
The $C^{1+α}$ hypothesis in Pesin theory revisited
摘要：
我们证明了对于每一个紧致的3-流形$M$，存在一个开的$\diff ^1(M)$子集，其中每一个类属同胚允许不可数的遍历概率测度，这些测度是双曲的，而它们的支集是不相交的，并且允许一个吸引邻域的基和一个排斥邻域的基.因此，支持这些测度的点既没有稳定流形，也没有非稳定流形。这与更高的正则性的情况下，在那里Pesin理论给我们的稳定和非稳定流形与互补的尺寸几乎在每一个点。我们还在第二维度上给出了这样的例子，没有局部泛化性。
英文摘要：
We show that for every compact 3-manifold $M$ there exists an open subset of $\diff ^1(M)$ in which every generic diffeomorphism admits uncountably many ergodic probability measures which are hyperbolic while their supports are disjoint and admit a basis of attracting neighborhoods and a basis of repelling neighborhoods. As a consequence, the points in the support of these measures have no stable and no unstable manifolds. This contrasts with the higher regularity case, where Pesin theory gives us the stable and the unstable manifolds with complementary dimensions at almost every point. We also give such an example in dimension two, without local genericity.
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