Can points of bounded orbits surround points of unbounded orbits ?

We show a somewhat surprising result: if E is a disk in the plane R², then there is a homeomorphism h: R²→ R² such that, for every x∈∂ E, the orbit $O(x, h)$ is bounded, but for every y∈ Int(E), the orbit $O(y, h)$ is doubly divergent. To prove this, we define a class of homeomorphisms on the square [-1, 1]², called normally rising homeomorphisms, and show that a normally rising homeomorphism can have very complex ω-limit sets and α-limt sets, though the homeomorphism itself looks very simple.