The groupoid C*-algebra of a rational map

TL;DR

This paper provides a detailed analysis of the C*-algebra associated with a rational map on the Riemann sphere, exploring its structure through extensions influenced by the map's Julia and Fatou sets.

Contribution

It offers a novel stepwise decomposition of the C*-algebra based on the dynamics of the rational map and the structure of its Julia and Fatou sets.

Findings

Decomposition of the C*-algebra into extensions of familiar C*-algebras.

Dependence of algebra structure on Julia set and Fatou set regions.

Analysis of critical points' influence on the algebra.

Abstract

This paper contains a quite detailed description of the C*-algebra arising from the transformation groupoid of a rational map of degree at least two on the Riemann sphere. The algebra is decomposed stepwise via extensions of familiar C*-algebras whose nature depend on the structure of the Julia set and the stable regions in the Fatou set, as well as on the behaviour of the critical points.