MINIMAL BUT NOT UNIQUELY ERGODIC DIFFEOMORPHISMS
MINIMAL BUT NOT UNIQUELY ERGODIC DIFFEOMORPHISMS
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最小但不是唯一的遍历微分态
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A. Windsor
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作者：
A. Windsor
如果f是可分度量空间上保持Borel测度μ的唯一遍历变换，则f|supp(μ)是极小的，即每个轨道是稠密的[7，命题4.1.18]。因此，最小限度是否足以实现独特的遍历，这是一个自然的问题。这是一系列问题中的一个，这些问题询问哪些拓扑性质对于更抽象的度量性质是足够的。不幸的答案是，尽管度量属性具有强大的拓扑结果，但反之亦然。其中一个原因是存在光滑上循环，这些上循环是可测的上边界，但其传递函数在拓扑上表现得很疯狂。这是A.N.Kolmogorov[1]所知道的，他用这种方法构造了T上具有纯点谱但具有不连续特征函数的线性流的时变。H.Furstenberg[3]独立地进行了同样的观察，他用它来构造极小但不是唯一遍历的微分同胚(关于构造的本质，见[7，推论12.6.4])。这些微分同胚是斜积的，是解析的，并且都允许无数的遍历度量。在符号动力学的情况下，S.Williams[6]给出了具有任意给定的有限数、可数数或遍历不变测度的连续体的极小变换的构造。这是我们前面关于符号系统的问题的一个非常普遍的反例。对于平滑类别，还没有发布这样的构造。D.V.Anosov和A.B.Katok[4]在没有证明的情况下，提出了一种拓扑传递微分同胚的构造，它保持光滑测度，且具有给定数目的遍历分支。我们使用他们的方法的一种变体来构造具有给定数目的遍历测度的极小微分同胚。横截是避免文献[4]中出现的度量零例外集的关键之一，A.B.Katok在1998年宾夕法尼亚州立大学的光滑遍历理论讲座中向作者传达了横割的思想。
If f is a uniquely ergodic transformation on a separable metric space preserving a Borel measure μ then f |supp(μ) is minimal, that is every orbit is dense [7, Proposition 4.1.18]. It was therefore a natural question whether minimality is sufficient for unique ergodicity. This is one in a series of questions which ask which topological properties are sufficient for more abstract metric properties. The unfortunate answer has been that while metric properties have strong topological consequences the converse is false. One reason for this is that there exist smooth cocycles which are measurable coboundaries but for which the transfer functions behave wildly topologically. This was known by A. N. Kolmogorov [1] who used this method to construct a time change of a linear flow on T with pure point spectrum but with discontinuous eigenfunctions. The same observation was independently made by H. Furstenberg [3] who used it to construct diffeomorphisms which are minimal but not uniquely ergodic (see [7, Corollary 12.6.4] for the essence of the construction). These diffeomorphisms are skew-products, are analytic, and all admit uncountably many ergodic measures. A construction of minimal transformations with any given finite number, a countable number, or a continuum of ergodic invariant measures was provided by S. Williams [6] in the case of symbolic dynamics. This serves as very general counterexample to our earlier question in the case of symbolic systems. No such construction has been published for the smooth category. A construction of topologically transitive diffeomorphisms preserving a smooth measure and with a given number of ergodic components was announced by D. V. Anosov and A. B. Katok [4] without proof. We use a variation of their methods to construct minimal diffeomorphisms with a given number of ergodic measures. The idea of transversal cutting which is one of the keys for avoiding the measure zero exceptional set which occurs in [4] was communicated to the author by A.B Katok in lectures on smooth ergodic theory given at The Pennsylvania State University during 1998.