C* -algebras associated with complex dynamical systems

Tsuyoshi Kajiwara, Yasuo Watatani

Research output: Contribution to journalArticle

21 Citations (Scopus)

Abstract

Iteration of a rational function R gives a complex dynamical system on the Riemann sphere. We introduce a C*-algebra script O signR associated with R as a Cuntz-Pimsner algebra of a Hubert bimodule over the algebra A = C(JR) of continuous functions on the Julia set J R of R. The algebra OR is a certain analog of the crossed product by a boundary action. We show that if the degree of R is at least two, then C* -algebra OR is simple and purely infinite. For example if R(z) = z2 - 2, then the Julia set JR = [-2,2] and the restriction R: JR → JR is topologically conjugate to the tent map on [0,1]. The algebra OZ2-2 is isomorphic to the Cuntz algebra O∞. We also show that the Lyubich measure associated with R gives a unique KMS state on the C*-algebra OR for the gauge action at inverse temperature log(deg-R), if the Julia set contains no critical points. Indiana University Mathematics Journal

Original languageEnglish
Pages (from-to)755-778
Number of pages24
JournalIndiana University Mathematics Journal
Volume54
Issue number3
DOIs
Publication statusPublished - Aug 16 2005

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Keywords

  • C*-algebras
  • Complex dynamical systems
  • Cuntz-Pimsner algebras
  • Rational functions

ASJC Scopus subject areas

  • Mathematics(all)

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