Measures and topological dynamics on Menger manifolds

We study nonatomic, locally positive, Lebesgue–Stieltjes measures on compact Menger manifolds and show that the set of all ergodic homeomorphisms on any compact Menger manifold X forms a dense Gδ set in the space of all measure preserving autohomeomorphisms of X with the compact-open topology. In particular, there exists a topologically transitive homeomorphism on any compact Menger manifold, which answers a question posed by several authors.We also prove the existence of homeomorphisms that are chaotic in the sense of Devaney as well as everywhere chaotic in the sense of Li–Yorke.