Simultaneous uniformization for uniformly quasisymmetric circle dynamical systems

作者：Gardiner Frederick P; Jiang Yunping^*

来源：Dynamical Systems, 2015, 30(4): 543-565.

DOI：10.1080/14689367.2015.1082537

• simultaneous uniformization
• uniformly quasisymmetric circle endomorphism
• uniformly symmetric circle endomorphism
• normalized quasidisc pairing
• conformal welding
• dynamical welding

In the 1960s Bers showed how to uniformize simultaneously two Riemann surfaces of the same finite analytic type by using a single quasi-Fuchsian group of the first kind. In this paper, we show how to uniformize simultaneously two uniformly quasisymmetric circle endomorphisms of the same degree by a unique normalized branched covering of the Riemann sphere of the same degree such that this branched covering has a unique normalized quasicircle as an invariant limit set. We use this simultaneous uniformization to define a transformation between their spaces of probability invariant measures and formulate several equivalent conjectures to the uniqueness conjecture for symmetric invariant probability measures. In a subsequent paper, we study these conjectures.

• 出版日期2015-10-2