By Branner B., Fagella N.
ISBN-10: 1107042917
ISBN-13: 9781107042919
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Extra info for Quasiconformal Surgery in Holomorphic Dynamics
Sample text
Indeed, Eu1 is defined for all u such that Ef puq is defined and Du f exists and is non-singular. The first property is satisfied on a set of full measure because f is absolutely continuous and so is the second property by hypothesis. 7 The pullback of the Beltrami coefficient μ (in V ) under f is denoted by f ˚ μ (in U ). When we write f pU, μ1 q ÝÑ pV , μ2 q we mean that f : U Ñ V and f ˚ μ2 “ μ1 , in the sense explained above. e. μ “ μg ), then we will be looking at f ˚ μg “ f ˚ pg ˚ μ0 q “ pg ˝ f q˚ μ0 “ μg ˝f .
28 (Integrability Theorem – global version) Let S be a simply connected Riemann surface and σ be an almost complex structure on S with measurable Beltrami form μ. e. Kpσ q ă 8 or, equivalently, the essential supremum of |μ| on S is }μ}8 “ k ă 1. e. there exists a quasiconformal homeomorphism φ : p which satisfies S Ñ D (respectively onto C or C) φ ˚ μ0 “ μ. p then φ : S Ñ D (respectively If S is isomorphic to D (respectively to C or C) p onto C or C) is unique up to post-composition with automorphisms of D p (respectively of C or C).
16) r is exactly and twice the argument from the real axis to the minor axis of E zq: equal to the argument of μϕrpr zq. 17) r represents the infinitesimal ellipse defined by the Therefore, the ellipse E zq. It follows that the measurable field of infinitesimal Beltrami coefficient μϕrpr ellipses in T S is well defined and, hence, that the Beltrami form with bounded dilatation determines an almost complex structure on S. The last thing we define is the pullback of a Beltrami form by a quasiconformal mapping.
Quasiconformal Surgery in Holomorphic Dynamics by Branner B., Fagella N.
by Ronald
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