C* -algebras associated with complex dynamical systems
C* -algebras associated with complex dynamical systems
复制标题
DOI：
10.1512/iumj.2005.54.2530
复制
发表时间：
2003-09
影响因子：
1.1
通讯作者：
Tsuyoshi Kajiwara;Y. Watatani
中科院分区：
文献类型：
--
作者：
Tsuyoshi Kajiwara;Y. Watatani
一个有理函数R的迭代给出了黎曼球上的一个复杂动力系统。在Julia集合jr (R)上的连续函数的代数a = C（jr）上，引入与R相关的C*-代数OR作为Hilbert双模的Cuntz-Pimsner代数。代数OR是由边界作用的交叉积的某种类比。我们证明了如果R的阶数至少为2，则C*-代数OR是简单且纯粹无限的。例如，若R(z) = z2 -2，则Julia集合jr =[-2,2]，且约束R: jr→jr拓扑共轭于[0,1]上的帐篷映射。代数oz2 -2与孔兹代数O∞同构。我们还证明，如果Julia集合不包含临界点，则与R相关的柳比奇测度在C*-代数或上给出了测量作用在逆温度对数（度R）下的唯一KMS状态。
Iteration of a rational function R gives a complex dynamical system on the Riemann sphere. We introduce a C*-algebra OR associated with R as a Cuntz-Pimsner algebra of a Hilbert bimodule over the algebra A = C(J R ) 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) = z 2 - 2, then the Julia set J R = [-2,2] and the restriction R: J R → J R is topologically conjugate to the tent map on [0,1]. The algebra O z2-2 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.