By Whyburn G.T.

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Suppose that f : D → D is a homeomorphism where D and D are domains in Rn . 9 Analytic Definition for Quasiconformality 33 for each family Γ of paths in D. 6. e. differentiable (in fact, it suffices that f is an open map). 7. e. differentiable homeomorphism and u ≥ 0 is a measurable function in D . Then u( f (x)) |J(x, f )| dx ≤ D u dy. 20. 32), one has to assume that f is ACLn . For a more detailed discussion; see [193]. 12. 31), fix a family Γ of paths in D. Since f is ACLn , the Fuglede theorem implies that f (the coordinate functions of f ) is absolutely continuous on a path family Γ0 of n-almost all paths in Γ .

More quantitatively, if f : X → Y is η -quasisymmetric and if A ⊂ B ⊂ X are such that 0 < diam A ≤ diam B < ∞, then diam f (B) is finite and 2η diam B diam A −1 ≤ diam f (A) ≤η diam f (B) 2 diam A . 43) Doubling spaces. Quasisymmetry is intimately connected to a property of a metric space called a doubling property. A metric space is called doubling if there is a constant C1 ≥ 1 so that every set of diameter d in the space can be covered by at most C1 sets of diameter at most d/2. It is clear that subsets of doubling spaces are doubling.

If n = 2 and K = 1, this definition leads to one of the most general definitions of analytic functions. 29). 11. e. x ∈ D, (c) f is sense-preserving. , f is quasiconformal according to the analytic definition. 18. The property that a mapping f : D → Rn is sense-preserving can be defined for every continuous mapping f with the aid of the topological degree; cf. 32 2 Moduli and Capacity [246]. e. 11. e. differentiable and ACLn . e. 29). If f is differentiable at x and J(x, f ) = 0, then f (x) = 0 because H(x, f ) ≤ c.