Title: Compact Group Homeomorphisms Preserving The Haar Measure Show Chinese title

Title: 紧致群自同胚保持哈尔测度

Abstract: This paper studies the measure-preserving homeomorphisms on compact groups and proposes new methods for constructing measure-preserving homeomorphisms on direct products of compact groups and non-commutative compact groups. On the direct product of compact groups, we construct measure-preserving homeomorphisms using the method of integration. In particular, by applying this method to the \(n\)-dimensional torus \({\mathbb{T}}^{n}\), we can construct many new examples of measure-preserving homeomorphisms. We completely characterize the measure-preserving homeomorphisms on the two-dimensional torus where one coordinate is a translation depending on the other coordinate, and generalize this result to the \(n\)-dimensional torus. For non-commutative compact groups, we generalize the concept of the normalizer subgroup \(N\left( H\right)\) of the subgroup \(H\) to the normalizer subset \({E}_{K}( P)\) from the subset \(K\) to the subset \(P\) of the group of measure-preserving homeomorphisms. We prove that if \(\mu\) is the unique \(K\)-invariant measure, then the elements in \({E}_{K}\left( P\right)\) also preserve \(\mu\). In some non-commutative compact groups the normalizer subset \({E}_{G}\left( {\mathrm{{AF}}\left( G\right) }\right)\) can give non-affine homeomorphisms that preserve the Haar measure. Finally, we prove that when \(G\) is a finite cyclic group and a \(n\)-dimensional torus, then \(\mathrm{{AF}}\left( G\right)= N\left( G\right) = {E}_{G}\left( {\mathrm{{AF}}\left( G\right) }\right)\). Show Chinese abstract

Abstract: 本文研究了紧致群上的保度量同胚，并提出了在紧致群的直积以及非交换紧致群上构造保度量同胚的新方法。在紧致群的直积上，我们利用积分方法构造保度量同胚。特别是，将这种方法应用于\(n\)维环面\({\mathbb{T}}^{n}\)时，我们可以构造许多新的保度量同胚的例子。我们完全刻画了二维环面上保度量同胚的情况，其中一坐标是依赖于另一坐标的平移，并将此结果推广到\(n\)维环面。 对于非交换紧致群，我们将子群 \(N\left( H\right)\) 的正规子群概念推广到保测同胚群中从子集 \(K\) 到子集 \(P\) 的正规子集 \({E}_{K}( P)\)，其中 \(H\) 是一个子群。 我们证明如果\(\mu\)是唯一的\(K\)不变量测度，则\({E}_{K}\left( P\right)\)中的元素也保持\(\mu\)。在某些非交换紧致群中，正规子集\({E}_{G}\left( {\mathrm{{AF}}\left( G\right) }\right)\)可以给出保持哈尔测度的非仿射同胚。 最后，我们证明当\(G\)是一个有限循环群且是一个\(n\)维环面时，则有\(\mathrm{{AF}}\left( G\right)= N\left( G\right) = {E}_{G}\left( {\mathrm{{AF}}\left( G\right) }\right)\)。