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

Abstract

Let K be an algebraically closed field which is complete with respect to a non-trivial non-Archimedean absolute value |⋅|. We study metric properties of the limit set Λ of a semigroup G generated by a finite set of contractive analytic functions on O={z∈K||z|⩽1}. We prove that the limit set Λ 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 Qp of p-adic numbers, or the limit set Λ satisfies the strong open set condition, then Λ has the doubling property. This yields that the limit set Λ is quasisymmetrically equivalent to the space Z2 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 Λ, and we prove that the limit set Λ has a positive capacity in the Berkovich space which yields that there exists an equilibrium measure μ whose support is contained in the limit set Λ. We also show that if the semigroup is generated by a countable set of contractive analytic functions, then its limit set Λ can be non-compact. However, if coefficients of the generating functions lie in Qp, then the limit set Λ is compact.