A conjugated version of the countable dense homogeneity

We are interested in finding a homeomorphism h of a space X with h −1 ○ Φ ○ h ( A ) = B for a given bijection Φ of X and every pair of countable dense subsets A and B of X. For a separable Banach space X, such a homeomorphism h always exists provided the fixed-point set of Φ has the empty interior. Moreover, h can be chosen to be real-analytic. As a consequence, there exists a real analytic flow that sends A onto B after time t = 1 . Actually, for X = R n , any bounded real-analytic vector field can be approximated by a real-analytic vector field whose induced flow sends A onto B after time t = 1 . Topological and C p smooth counterparts of these results are also obtained.