电子隐形的级联
Cascades of e-invisibility
摘要：
我们考虑局部典型动力系统的统计吸引子和它们的“e-不可见”子集：其邻域被平均频率小于e << 1的轨道访问的吸引子的一部分。对于非常小的e值（比如说，小于2^（-10^6）），观察者在遵循一般轨道时几乎看不到这些部分。一般动力系统中e-不可见性的一个微不足道的原因可能是映射的高Lipshitz常数（~1/e）（即它严重扭曲了度量）或它与结构不稳定的动力系统的接近度（~e）。然而，Ilyashenko和Negut [IN]提供了一个局部典型的例子，动力系统的e-不可见集和一个均匀的中等（<100）Lipshitz常数独立于e。这些来自[IN]的动力系统也是|log e| ^{-1}-远离结构不稳定的动力系统（在斜积的S类中）。我们进一步发展[IN]的例子，以提供更好的不可见率，同时保持在相同的距离远离结构不稳定的动力系统。我们给出了一个明确的例子，C^1-球在伯努利移位上的“阶”斜积空间中，使得对于来自这个球的每个动力系统，统计吸引子的大部分是不可见的。与结构不稳定系统（在S类中）相距c/n-距离的系统的不可见率e = 2^（-n^k），其中3 k是相空间的豪斯多夫维数。
英文摘要：
We consider statistical attractors of locally typical dynamical systems and their "e-invisible" subsets: parts of the attractors whose neighborhoods are visited by orbits with an average frequency of less than e << 1. For extraordinarily small values of e (say, smaller than 2^(-10^6)), an observer virtually never sees these parts when following a generic orbit. A trivial reason for e-invisibility in a generic dynamical system may be either a high Lipshitz constant (~1/e) of the mapping (i.e. it badly distorts the metric) or its proximity (~e) to the structurally unstable dynamical systems. However Ilyashenko and Negut [IN] provided a locally typical example of dynamical systems with an e-invisible set and a uniform moderate (<100) Lipshitz constant independent on e. These dynamical systems from [IN] are also |log e|^{-1}-distant from structurally unstable dynamical systems (in the class S of skew products). We further develop the example of [IN] to provide a better rate of invisibility while staying at the same distance away from the structurally unstable dynamical systems. We give an explicit example of C^1-balls in the space of "step" skew products over the Bernoulli shift such that for each dynamical system from this ball a large portion of the statistical attractor is invisible. Systems that are c/n-distant from structurally unstable ones (in the class S) have rate of invisibility e = 2^(-n^k) where 3k is the Hausdorff dimension of the phase space.
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