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

2012/09/05 by Tsuyoshi Kajiwara, Kajiwara, Tsuyoshi, Yasuo Watatani +1
Mathematics · Physics and Astronomy · #46L08 #46L55 #Advanced Operator Algebra Research #Dynamical Systems (math.DS) #FOS: Mathematics #Noncommutative and Quantum Gravity Theories #Operator Algebras (math.OA) #Spectral Theory in Mathematical Physics #math.DS #math.OA #msc:46L08 #msc:46L55

paper · pdf · doi:10.48550/arxiv.1209.0850

12 pages

arxiv created 2012/09/05 · openalex publication_date 2012/09/05 · arxiv updated 2012/09/06 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

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.

Citations

• Ideal structure and simplicity of the C*-algebras generated by Hilbert bimodules

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