Compact Group Homeomorphisms Preserving The Haar Measure

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 $E_G\left(\mathrm{A}\mathrm{F}\left(G\right)\right)$ 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 $\mathrm{A}\mathrm{F}\left(G\right)=N\left(G\right)=E_G\left(\mathrm{A}\mathrm{F}\left(G\right)\right)$.