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We construct measure-preserving mappings from the $d$-dimensional unit cubeto the $d$-dimensional unit ball and the compact rank one symmetric spaces,namely the $d$-dimensional sphere, the real, complex, and quaternionicprojective spaces, and the Cayley plane. We also give a procedure to generatemeasure-preserving mappings from the $d$-dimensional unit cube to productspaces and fiber bundles under certain conditions.

Measure-preserving mappings from the unit cube to some symmetric spaces