Escaping sets of continuous functions

TL;DR

This paper investigates which subsets of Euclidean space can be realized as escaping sets of continuous functions, revealing differences between one-dimensional and higher-dimensional cases and highlighting the problem's complexity.

Contribution

The paper provides partial characterizations of escaping sets in various dimensions, especially emphasizing the distinct behaviors in one dimension versus higher dimensions.

Findings

In one dimension, the structure of escaping sets differs significantly from higher dimensions.

Open sets can be realized as escaping sets under certain conditions.

The problem exhibits surprising complexity and richness across different cases.

Abstract

Our objective is to determine which subsets of $\mathbb{R}^d$ arise as escaping sets of continuous functions from $\mathbb{R}^d$ to itself. We obtain partial answers to this problem, particularly in one dimension, and in the case of open sets. We give a number of examples to show that the situation in one dimension is quite different from the situation in higher dimensions. Our results demonstrate that this problem is both interesting and perhaps surprisingly complicated.