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Factoring the logarithmic spiral. (English) Zbl 0622.30011

If f is a K-quasiconformal homeomorphism of Jordan domains in the plane, then for any given \(c>1\), f is equal to some composition of \(N<\log _ cK+1\) c-quasiconformal homeomorphisms [see e.g. O. Lehto, Univalent functions and Teichmüller spaces (1987; Zbl 0606.30001), Theorem 4.7]. It is not known if maps of large distortion K can always be factored into a composition of maps of smaller distortion c when ”quasiconformal” is replaced by ”quasi-isometric”. We establish that such factorizations may require many more (asymptotically K/c) factors in the quasi-isometric case. Consider the map \(s_ k: \bar D^ 2=\{(x_ 1,x_ 2)\in {\mathbb{R}}^ 2\); \(x^ 2_ 1+x^ 2_ 2\leq 1\}\to \bar D^ 2\) defined in polar coordinates by: \[ s_ k(\rho,\theta)=(\rho,\theta +k \log \rho), \] where \(k=L-(1/L)\), \(L>1\). Then it is easy to show that \(s_ k\) is an L-quasi-isometry, i.e., \[ (1/L)| p-q| \leq | s_ k(p)-s_ k(q)| \leq L| p-q|,\quad \forall p,q\in \bar D^ 2. \] The main result of the paper shows that it requires \(N\geq k/\sqrt{\alpha ^ 2-1}\) (rather than \(\log _{\alpha}k+1)\) factors to write \(s_ k\) into a composition of \(\alpha\)-quasi-isometries. The conformal structure of planar annuli is used to define a notion of ”twist” for a homeomorphism. The proof analyses how a given amount of twist can be shared among factors.

MSC:

30C62 Quasiconformal mappings in the complex plane

Citations:

Zbl 0606.30001
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References:

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