C*-algebras associated with complex dynamical systems and backward orbit structure

TL;DR

This paper explores the connection between $C^*$-algebras and complex dynamical systems generated by rational functions, revealing how algebraic structures encode the behavior of backward orbits at branched points.

Contribution

It introduces a method to recover the count of backward orbits at branched points using $C^*$-algebras with gauge actions, linking algebraic and dynamical properties.

Findings

Count of backward orbits at branched points expressed via $C^*$-algebras

Partial understanding of how branched points move under iteration

Use of KMS states and Perron-Frobenius operators to analyze dynamics

Abstract

Let $R$ be a rational function. The iterations $(R^n)_n$ of $R$ gives a complex dynamical system on the Riemann sphere. We associate a $C^*$-algebra and study a relation between the $C^*$-algebra and the original complex dynamical system. In this short note, we recover the number of $n$-th backward orbits counted without multiplicity starting at branched points in terms of associated $C^*$-algebras with gauge actions. In particular, we can partially imagine how a branched point is moved to another branched point under the iteration of $R$. We use KMS states and a Perron-Frobenius type operator on the space of traces to show it.