The Dynamics of Twisted Tent Maps

TL;DR

This paper explores the complex dynamics of twisted tent maps, a new family of functions that combine complex scaling and folding, revealing behaviors from geometric to chaotic fractals.

Contribution

Introduces and analyzes the dynamics of twisted tent maps, a novel complex generalization of real tent maps with unique folding and twisting behaviors.

Findings

Maps exhibit a range of behaviors from geometric to chaotic fractals.

Folding breaks analyticity and smoothness, leading to complex dynamics.

The maps demonstrate rich structures including fractals and chaotic attractors.

Abstract

This paper is a study of the dynamics of a new family of maps from the complex plane to itself, which we call twisted tent maps. A twisted tent map is a complex generalization of a real tent map. The action of this map can be visualized as the complex scaling of the plane followed by folding the plane once. Most of the time, scaling by a complex number will "twist" the plane, hence the name. The "folding" both breaks analyticity (and even smoothness) and leads to interesting dynamics ranging from easily understood and highly geometric behavior to chaotic behavior and fractals.