Chaotic dynamics of a quasiregular sine mapping

TL;DR

This paper explores the chaotic behavior of a quasiregular sine-like map in higher dimensions, demonstrating dense periodic points and that the Julia set covers the entire space, indicating complex dynamics.

Contribution

It establishes the density of periodic points and the nature of the Julia set for a quasiregular sine mapping, extending complex dynamics concepts to higher dimensions.

Findings

Periodic points are dense in .

The Julia set equals , indicating chaotic dynamics.

The map's escaping set is dense in .

Abstract

This article studies the iterative behaviour of a quasiregular mapping S:\R^d\to\R^d that is an analogue of a sine function. We prove that the periodic points of S form a dense subset of \R^d. We also show that the Julia set of this map is \R^d in the sense that the forward orbit under S of any non-empty open set is the whole space \R^d. The map S was constructed by Bergweiler and Eremenko who proved that the escaping set I(S) is also dense in \R^d.