The K-theory of the C*-algebras associated to rational functions

TL;DR

This paper computes the K-theory of C*-algebras linked to rational functions on the Riemann sphere, revealing new dynamical invariants and connecting algebraic properties with complex dynamics.

Contribution

It introduces new exact sequences in K-theory for C*-correspondences and classifies the associated C*-algebras, providing a novel algebraic perspective on rational function dynamics.

Findings

K-theory depends only on degree, critical points, and Fatou cycles of R

The Julia set C*-algebra is a unital UCT Kirchberg algebra

Results offer new invariants and relate to the Density of Hyperbolicity Conjecture

Abstract

We compute the K-theory of the three C*-algebras associated to a rational function R acting on the Riemann sphere, its Fatou set, and its Julia set. The latter C*-algebra is a unital UCT Kirchberg algebra and is thus classified by its K-theory. The K-theory in all three cases is shown to depend only on the degree of R, the critical points of R, and the Fatou cycles of R. Our results yield new dynamical invariants for rational functions and a C*-algebraic interpretation of the Density of Hyperbolicity Conjecture for quadratic polynomials. These calculations are possible due to new exact sequences in K-theory we induce from morphisms of C*-correspondences.