Compact Group Homeomorphisms Preserving The Haar Measure

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)).