Can points of bounded orbits surround points of unbounded orbits ?

Abstract

We show a somewhat surprising result: if $E$ is a disk in the plane $\mathbb R^2$, then there is a homeomorphism $h:\mathbb R^2\rightarrow\mathbb R^2$ such that, for every $x\in\partial E$, the orbit $O(x, h)$ is bounded, but for every $y\in {\rm Int}(E)$, 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 $\omega$-limit sets and $\alpha$-limt sets, though the homeomorphism itself looks very simple.