Hausdor dimension and conformal dynamics I: Strong convergence of Kleinian groups
نویسنده
چکیده
This paper investigates the behavior of the Hausdorr dimensions of the limit sets n and of a sequence of Kleinian groups ? n ! ?, where M = H 3 =? is geometrically nite. We show if ? n ! ? strongly, then: (a) M n = H 3 =? n is geometrically nite for all n 0, (b) n ! in the Hausdorr topology, and (c) H: dim((n) ! H: dim((), if H: dim(() 1. On the other hand, we give examples showing the dimension can vary discontinuously under strong limits when H: dim(() < 1. Continuity can be recovered by requiring that accidental parabolics converge radially. Similar results hold for higher-dimensional manifolds. Applications are given to quasifuchsian groups and their limits.
منابع مشابه
Hausdorff dimension and conformal dynamics I: Strong convergence of Kleinian groups
This paper investigates the behavior of the Hausdorff dimensions of the limit sets Λn and Λ of a sequence of Kleinian groups Γn → Γ, where M = H/Γ is geometrically finite. We show if Γn → Γ strongly, then: (a) Mn = H 3/Γn is geometrically finite for all n ≫ 0, (b) Λn → Λ in the Hausdorff topology, and (c) H. dim(Λn) → H. dim(Λ), if H. dim(Λ) ≥ 1. On the other hand, we give examples showing the ...
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