Abstract
Distortion of length and angle are widely studied in geometry and dynamics. Much of the success in the study of one-dimensional complex dynamics flow from the Ahlfors-Bers-Sullivan [A, B] [S 1] theory for deforming conformal structures essentially from the existence and uniqueness theory for the Beltrami equation d~f/d~f=#. No comparable analytic theory exists for the distortion of length in the plane or of length or angle in higher dimensions. (In these cases the local distortion is specified by more than d parameters in dimension = d, so the analogous equations are over determined.) What exists instead is a coarser, more topological, discussion initiated by Sullivan of structures, extension and isotopy of imbeddings [S 2], [T, V] in the quasiconformal and bilipschitz (-quasi-isometric) categories. The "coarseness" is the use of the Alexander isotopy which crushes rather than unwinds homeomorphisms. In this paper we find a constraint on the more delicate method of returning a quasi-isometry to the identity: factorization into maps of small (isometric) distortion. We study the following L-quasi-isometry of the closed unit disk/~a = {(xl, x2); x 2 + x2 < 1 } :
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