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Quasiconformal homeomorphisms in dynamics, topology, and geometry. (English) Zbl 0698.58030

Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 2, 1216-1228 (1987).
[For the entire collection see Zbl 0657.00005.]
This congress paper first defines the notion of a quasiconformal homeomorphism between arbitrary metric spaces. This phenomenon is then analyzed in 4 different situations. First a distance between real analytic dynamical systems is defined using quasiconformal conjugacies and shown to be decreased by Feigenbaum’s renormalization operator. Next it is shown that each topological manifold has an essentially unique quasiconformal structure - with the famous exception of dimension 4, well-known from differential structures. In the third part a complex analytic classification of expanding transformations near fractal invariant sets is discussed. In the last part of the paper a characterization of constant curvature among variable negative curvature in terms of measure theoretical dynamical properties is given. This turns out to be equivalent to uniform quasiconformality of a geodesic flow.
Reviewer: C.H.Cap

MSC:

37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
37B99 Topological dynamics
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
53C20 Global Riemannian geometry, including pinching
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)

Citations:

Zbl 0657.00005