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

Fingerprint

Complex Dynamical Systems
Julia set
C*-algebra
Cuntz Algebra
Algebra
KMS States
Tent Map
Crossed Product
Bimodule
Rational function
Critical point
Gauge
Continuous Function
Isomorphic
Restriction
Analogue
Iteration

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

C* -algebras associated with complex dynamical systems. / Kajiwara, Tsuyoshi; Watatani, Yasuo.

In: Indiana University Mathematics Journal, Vol. 54, No. 3, 16.08.2005, p. 755-778.

Research output: Contribution to journalArticle

Kajiwara, Tsuyoshi ; Watatani, Yasuo. / C* -algebras associated with complex dynamical systems. In: Indiana University Mathematics Journal. 2005 ; Vol. 54, No. 3. pp. 755-778.
@article{9b0afd06e2e44d9e865825e088b1af85,
title = "C* -algebras associated with complex dynamical systems",
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",
author = "Tsuyoshi Kajiwara and Yasuo Watatani",
year = "2005",
month = "8",
day = "16",
doi = "10.1512/iumj.2005.54.2530",
language = "English",
volume = "54",
pages = "755--778",
journal = "Indiana University Mathematics Journal",
issn = "0022-2518",
publisher = "Indiana University",
number = "3",

}

TY - JOUR

T1 - C* -algebras associated with complex dynamical systems

AU - Kajiwara, Tsuyoshi

AU - Watatani, Yasuo

PY - 2005/8/16

Y1 - 2005/8/16

N2 - 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

AB - 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

UR - http://www.scopus.com/inward/record.url?scp=23244457914&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=23244457914&partnerID=8YFLogxK

U2 - 10.1512/iumj.2005.54.2530

DO - 10.1512/iumj.2005.54.2530

M3 - Article

AN - SCOPUS:23244457914

VL - 54

SP - 755

EP - 778

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

SN - 0022-2518

IS - 3

ER -