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 $h:{\mathbb{R}}^2\to {\mathbb{R}}^2$ such that, for every $x\in \partial E$, the orbit $O(x,h)$ is bounded, but for every $y\in \mathrm{Int}(E)$, 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 $\ddot{I}‰$-limit sets and $\hat{I}\pm$-limt sets, though the homeomorphism itself looks very simple.

18 pages. Comments are welcome