Almost Periodic Ergodic Rn-Homeomorphisms

Abstract

Listen

Translate article

A homeomorphism of Rn is called stationary if it is the uniform limit of volume preserving homeomorphisms which are spatially periodic and have mean translation zero. We prove that ergodic homeomorphisms form a uniform topology dense Gδ subset of the stationary homeomorphisms, thus establishing examples which are uniformly continuous with bounded and almost periodic displacements. In the two-dimensional orientation preserving case, ergodic homeomorphisms have fixed points (by Brouwer′s Plane Translation Theorem). Hence so do orientation-preserving area-preserving torus homeomorphisms with mean translation zero lifts (Conley-Zehnder-Frank Theorem).