Can points of bounded orbits surround points of unbounded orbits ?

TL;DR

The paper constructs a homeomorphism of the plane where boundary points have bounded orbits while interior points have doubly divergent orbits, revealing complex dynamical behaviors in simple homeomorphisms.

Contribution

It introduces normally rising homeomorphisms on the square and demonstrates their complex limit set structures, challenging intuitive expectations about orbit behaviors.

Findings

Boundary points have bounded orbits.

Interior points exhibit doubly divergent orbits.

Homeomorphisms can have complex limit sets despite simple forms.

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.