Self-similarity of Jankins-Neumann ziggurat

TL;DR

This paper investigates the self-similarity of the Jankins-Neumann ziggurat, a structure arising from topological and dynamical systems questions related to circle homeomorphisms and their rotation numbers.

Contribution

It proves the self-similarity of the ziggurat using a formula by Calegari and Walker and provides a concise proof of the equivalence between different descriptions of the ziggurat.

Findings

Established the self-similarity of the ziggurat.

Provided a short proof of equivalence between two descriptions.

Connected topological questions with dynamical systems theory.

Abstract

Primarily having emerged from a topological question, Jankins-Neumann ziggurat also appears in the theory of dynamical systems on the circle. It describes an answer to the following question: given the rotation numbers of two orientation-preserving circle homeomorphisms, what can be said about the rotation number of their composition? In this paper, we consider a formula, proved by Calegari and Walker, as definition of ziggurat. Using it, we establish its self-similarity. Also, we propose a short proof of equivalence between Calegary-Walker and Jankins-Neumann's descriptions of the ziggurat.