Topological entropy and secondary folding

TL;DR

This paper investigates the discrepancy between homologically forced topological entropy and actual entropy in chaotic systems, attributing the difference to secondary folding phenomena observed in physical and mathematical models.

Contribution

It introduces the concept of secondary folding as a cause for entropy gaps and provides rigorous evidence of its occurrence in toral linked twist maps.

Findings

Secondary folds cause entropy to exceed homological bounds.

Secondary folding occurs in physical stirring devices and mathematical models.

Rigorous proof of secondary folds in toral linked twist maps.

Abstract

A convenient measure of a map or flow's chaotic action is the topological entropy. In many cases, the entropy has a homological origin: it is forced by the topology of the space. For example, in simple toral maps, the topological entropy is exactly equal to the growth induced by the map on the fundamental group of the torus. However, in many situations the numerically-computed topological entropy is greater than the bound implied by this action. We associate this gap between the bound and the true entropy with 'secondary folding': material lines undergo folding which is not homologically forced. We examine this phenomenon both for physical rod-stirring devices and toral linked twist maps, and show rigorously that for the latter secondary folds occur.