Universal bounds on the entropy of toroidal attractors

Abstract

A toroidal set is a compactum $K \subseteq \mathbb{R}^3$ which has a neighbourhood basis of solid tori. We study the topological entropy of toroidal attractors $K$, bounding it from below in terms of purely topological properties of $K$. In particular, we show that for a toroidal set $K$, either any smooth attracting dynamics on $K$ has an entropy at least $\log 2$, or (up to continuation) $K$ admits smooth attracting dynamics which are stationary (hence with a zero entropy).

Figures (2)

• Figure 1: Whitehead's continuum
• Figure 2: Setup for Lemma \ref{['lemma:curvesbound']}

Theorems & Definitions (50)

• Theorem 1.1
• Definition 2.1
• Example 2.2
• proof
• Remark 2.3
• Remark 2.4
• Definition 3.1
• Proposition 3.2
• proof
• Example 3.3