A conjugated version of the countable dense homogeneity
A separable metric space \(X\) is said to be countable dense homogeneous if for any two countable dense sets \(A\) and \(B\) in \(X\), there is a homeomorphism \(h: X\to X\) such that \(h(A)= B\). The author modifies this property considering a bijection \(\Phi: X\to X\) whose fixed point set has empty interior and such that given any two countable dense sets \(A\) and \(B\) in \(X\), there is a homeomorphism \(h: X\to X\) such that \(\Phi(h(A))= h(B)\). Among other results obtained he proves that if \(\Phi: E\to E\) is a bijection of a Banach space \(E\) such that \(\text{int}(\text{Fix}(\Phi))=\emptyset\), then, given any two countable dense sets \(A\) and \(B\) in \(E\), there exists a real-analytic diffeomorphism \(h: E\to E\) such that \(\Phi(h(A))= h(B)\). Note that \(h^{-1}(\Phi(h(A)))= B\) is equivalent to \(\Phi(h(A))= h(B)\).
scientific article; zbMATH DE number 5526277
scientific article