On metric properties of limit sets of contractive analytic non-Archimedean dynamical systems

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:414

Article

Qiu, Weiyuan¹  Wang, Yuefei²  Yang, Jinghua²  Yin, Yongchen³ 

[1] Fudan Univ, Sch Math Sci, Shanghai 200433, Peoples R China

[2] Chinese Acad Sci, Inst Math, AMSS, Beijing 100190, Peoples R China

[3] Zhejiang Univ, Dept Math, Hangzhou 310027, Zhejiang, Peoples R China

关键词: Contractive analytic functions;    Non-Archimedean spaces;    Uniform perfectness;    Berkovich space;    Limit sets;   

DOI  :  10.1016/j.jmaa.2014.01.015

来源: Elsevier

【 摘 要 】

Let K be an algebraically closed field which is complete with respect to a non-trivial non-Archimedean absolute value vertical bar.vertical bar We study metric properties of the limit set A of a semigroup G generated by a finite set of contractive analytic functions on O = {z is an element of K vertical bar vertical bar z vertical bar <= 1}. We prove that the limit set A of G is uniformly perfect if the derivative of each generating function of G does not vanish on O. Furthermore, we show that if each coefficient of the generating functions is in the field Q(p) of p-adic numbers, or the limit set A satisfies the strong open set condition, then A has the doubling property. This yields that the limit set A is quasisymmetrically equivalent to the space Z(2) of 2-adic integers. We also give a counterexample to show that not all limit sets have the doubling property. The Berkovich space is introduced to study the limit set A, and we prove that the limit set A has a positive capacity in the Berkovich space which yields that there exists an equilibrium measure mu whose support is contained in the limit set A. We also show that if the semigroup is generated by a countable set of contractive analytic functions, then its limit set A can be non-compact. However, if coefficients of the generating functions lie in Q(p), then the limit set A is compact. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

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