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Ahlfors annulus assume automorphisms Bers Bloch function bounded closed trajectories compact component cone condition conformal mapping conjugacy conjugate constant contains COROLLARY critical point curve cusp defined denote dilatation disjoint dynamics entire functions equation example exists extends fixed point Fuchsian group Gehring Hence holomorphic function homeomorphism homotopy identity implies inequality integral invariant isotopy iterates Jordan domain Julia set Kleinian groups Lemma meromorphic function metric Mobius transformation Mod(J,f modulus neighborhood orbifold parabolic periodic points plane Pommerenke proof of Theorem PROPOSITION proved punctured qr mapping quadratic differential quadrature domain quasicircle quasiconformal mappings quasisymmetric rational function rational map rel ideal boundary result Riemann sphere Riemann surface saddle connections satisfies self-maps simply connected solutions subgroup subset Suppose Teichmiiller Teichmuller space Theorem 4.1 theory topological uniformly unique unit disk univalent functions universal covering values wandering domain whole Riemann sphere zero