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The Fixed Points of the Transformation
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algebraic function analytic function angle automorphic with respect belonging boundary point bounded branch points cluster point congruent points congruent sides consider contains converges copies of F0 cross-cut curve cycle cyclic group domain of existence elliptic transformation exterior finite number finite plane fixed circle fixed points following theorem Fuchsian group fundamental region Hence inequality inner point integral interior of Q0 interior point inverse isometric circles joining Let zi limit points linear transformation loop-cut loxodromic lying mapped conformally mapping function neighborhood one-to-one ordinary point origin pair parabolic points plane region points congruent principal circle properly discontinuous radius rational function real axis Riemann surface rotation sequence sheets simple automorphic function simply connected single single-valued solutions square elements takes Theorem tion Tn(z trans transformation carrying triangle unit circle Q0 upper half plane vertex vertices whence whole plane z-plane