The topological complexity of Cantor attractors for unimodal interval maps

TL;DR

This paper investigates the topological complexity of Cantor attractors in unimodal interval maps, revealing a typical growth rate of order n log n, with special cases of bounded complexity constructed.

Contribution

It establishes the order of complexity growth for Cantor attractors in unimodal maps and provides explicit examples with bounded complexity.

Findings

Complexity function p(U, n) grows as n log n for typical Cantor attractors.

Constructs a non-renormalizable map with bounded complexity function.

Provides insights into the topological structure of Cantor attractors.

Abstract

For a non-flat $C^3$ unimodal map with a Cantor attractor, we show that for any open cover $\mathcal U$ of this attractor, the complexity function $p(\mathcal U, n)$ is of order $n\log n$. In the appendix, we construct a non-renormalizable map with a Cantor attractor for which $p(\mathcal{U}, n)$ is bounded from above for any open cover $\mathcal{U}$.