The conjugacy problem in ergodic theory
The conjugacy problem in ergodic theory
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遍历理论中的共轭问题
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
B. Weiss
中科院分区：
文献类型：
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作者：
M. Foreman;D. Rudolph;B. Weiss
所有常见的概率保持变换都可以表示为MPT的元素，MPT是具有勒贝格测度的单位区间的测度保持变换群。这个群具有自然的波兰拓扑，遍历变换集合上的诱导拓扑也是波兰拓扑。我们的主要结果是，MPT中与它们的逆同构的遍历元素T的集合是一个完全解析集。这就得出了同构关系也是一个完全解析集，特别是不是Borel的结论。这与用谱定理证明酉群中的共轭关系是Borel的情形形成了鲜明的对比。这个结果也许解释了为什么确定遍历变换是否同构的问题被证明是如此棘手。我们使用的构造足够普遍，足以证明具有非平凡中心化子的遍历T集合也是完全解析的。在积极的方面，我们证明了当约束于构成MPT的一般子集的秩一变换时，同构关系是Borel的。如何找到一种好的显式方法来检验两个秩一变换是否同构，仍然是一个有待解决的问题。纪念：在…之前
All common probability preserving transformations can be represented as elements of MPT, the group of measure preserving transformations of the unit interval with Lebesgue measure. This group has a natural Polish topology and the induced topology on the set of ergodic transformations is also Polish. Our main result is that the set of ergodic elements T in MPT that are isomorphic to their inverse is a complete analytic set. This has as a consequence the fact that the isomorphism relation is also a complete analytic set and in particular is not Borel. This is in stark contrast to the situation of unitary operators where the spectral theorem can be used to show that conjugacy relation in the unitary group is Borel. This result explains, perhaps, why the problem of determining whether ergodic transformations are isomorphic or not has proven to be so intractable. The construction that we use is general enough to show that the set of ergodic T ’s with nontrivial centralizer is also complete analytic. On the positive side we show that the isomorphism relation is Borel when restricted to the rank one transformations, which form a generic subset of MPT. It remains an open problem to nd a good explicit method of checking when two rank one transformations are isomorphic. In Memoriam: Prior to the