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, 1², called normally rising homeomorphisms, and show that a normally rising homeomorphism can have very complex -limit sets and -limt sets, though the homeomorphism itself looks very simple.