Can points of bounded orbits surround points of unbounded orbits ?

2024/04/21 by Mai, Jiehua, Shi, Enhui, Yan, Kesong +1
#37E30 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2404.13642

Abstract

We show a somewhat surprising result: if E is a disk in the plane \mathbb R², then there is a homeomorphism h:\mathbb R²→\mathbb R² such that, for every x∈∂ E, the orbit O(x, h) is bounded, but for every y∈ \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 ω-limit sets and α-limt sets, though the homeomorphism itself looks very simple.

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