C^*-Algebras Associated with Complex Dynamical Systems and Backward Orbit Structure

Let R be a rational function. The iterations (Rⁿ)_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.