Symmetric Structures on a Closed Curve

Abstract

We show the quasisymmetric topology of Ahlfors ([1], 1965) (the topology coming from uniform ratio distortion) on local homeomor? phisms in one real dimension is defined when, and only when, the underlying one-manifold is provided with a "symmetric structure," one defined by using as structure pseudogroup the quasisymmetric closure of the C-diffeomorphisms of the real line. We show that the set of all symmetric structures on a closed curve compatible with a background quasisymmetric structure is naturally a complete, complex Banach man? ifold, modelled on the Banach space A*/X*, where A* and X* are the spaces of continuous functions F on the circle introduced by Zygmund ([17], 1945); A* : F(x + t) 4-F(x-t)-2F(x) = 0(t) X* : F(x + 0 + F(x-t)-2F(x) = o(t) and the complex structure is given by the Hilbert transform. The discussion covers analytical and geometrical properties of sym? metric homeomorphisms and symmetric quasicircles and suggests how the Bers' embedding technique (1965) may be used in a variety of contexts. 0. Description of results. The notion of a quasisymmetric ho? meomorphism of an interval into U is useful in the theory of Riemann surfaces and, more generally, in the theory of one real dimensional smooth dynamical systems. The set of quasisymmetric homeomorphisms is closed under composition and inVerse and can be recognized locally

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