Critical intermittency in rational maps

TL;DR

This paper investigates critical intermittency in iterated rational maps on the Riemann sphere, focusing on the role of superattracting fixed points and their impact on dynamical properties like transitivity.

Contribution

It introduces specific examples of iterated rational maps exhibiting critical intermittency and analyzes their topological dynamics, highlighting new mechanisms involving superattracting fixed points.

Findings

Identification of conditions for critical intermittency in rational maps

Examples demonstrating superattracting fixed points leading to intermittency

Analysis of topological transitivity in these systems

Abstract

Intermittent dynamics is characterized by long periods of different types of dynamical characteristics, for instance almost periodic dynamics alternated by chaotic dynamics. Critical intermittency is intermittent dynamics that can occur in iterated function systems, and involves a superattracting periodic orbit. This paper will provide and study examples of iterated function systems by two rational maps on the Riemann sphere that give rise to critical intermittency. The main ingredient for this is a superattracting fixed point for one map that is mapped onto a common repelling fixed point by the other map. We include a study of topological properties such as topological transitivity.