Compact Group Homeomorphisms Preserving The Haar Measure

TL;DR

This work tackles the construction and structure of measure-preserving homeomorphisms on compact groups with respect to the Haar measure. It develops a coordinate-wise integration method to produce Haar-preserving maps on direct products (including explicit ${\mathbb{T}}^n$-torus examples) and provides a complete toral characterization of maps that preserve Haar measure, revealing that many higher-dimensional cases yield non-affine examples. The paper then generalizes the normalizer concept to generalized normalizers ${E}_{K}(P)$, proving that if a measure is unique invariant under $K$, elements of ${E}_{K}(P)$ preserve it, and showing that in non-commutative groups this can strictly expand the affine set (producing non-affine, Haar-preserving maps, e.g., on ${SO}(3)\times{SO}(3)$). Conversely, for commutative groups such as finite cyclic groups and tori, the normalizer collapses to the affine set, yielding a precise inclusion chain ${\{L_x\}}\subseteq {\mathrm{AF}}(G)\subseteq N({\mathrm{AF}}(G))\subseteq E_G({\mathrm{AF}}(G))$, which is sharp in these cases. Overall, the paper provides constructive tools and structural clarity for understanding the full group of Haar-measure-preserving homeomorphisms on compact groups, with concrete toral classifications and non-affine examples in the non-commutative setting.

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 \(μ\) is the unique \(K\)-invariant measure, then the elements in \({E}_{K}\left( P\right)\) also preserve \(μ\). 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 \(\mathrm{AF}\left( G\right)= N\left( G\right) = {E}_{G}\left( {\mathrm{AF}\left( G\right) }\right)\).