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Let f, g : S1 →S1 be two C3 critical homeomorphisms of the circle with the same irrational rotation number and the same (finite) number of critical points, all of which are assumed to be non-flat, of power-law type. In this paper we prove that if h : S1→S1 is a topological conjugacy between f and g and h maps the critical points of f to the critical points of g, then h is quasisymmetric. When the power-law exponents at all critical points are integers, this result is a special case of a general theorem recently proved by T. Clark and S. van Strien preprint, 2014. However, unlike their proof, which relies on heavy complex-analytic machinery, our proof uses purely realvariable methods and is valid for non-integer critical exponents as well. We do not require h to preserve the power-law exponents at corresponding critical points. [ABSTRACT FROM AUTHOR]

Let Ω be a planar Jordan domain. We consider double-dome-like surfaces Σ defined by graphs of functions of dist(., ∂Ω) over Ω. The goal is tofind the right conditions on the geometry of the base Ω and the growth of the height so that Σ is a quasisphere or is quasisymmetric to S2. An internal uniform chord-arc condition on the constant distance sets to ∂Ω, coupled with a mild growth condition on the height, gives a close-to-sharp answer. Our method also produces new examples of quasispheres in Rn, for any n ≥ 3. [ABSTRACT FROM AUTHOR]

Let Ω be a planar Jordan domain. We consider double-dome-like surfaces Σ defined by graphs of functions of dist(., ∂Ω) over Ω. The goal is tofind the right conditions on the geometry of the base Ω and the growth of the height so that Σ is a quasisphere or is quasisymmetric to S2. An internal uniform chord-arc condition on the constant distance sets to ∂Ω, coupled with a mild growth condition on the height, gives a close-to-sharp answer. Our method also produces new examples of quasispheres in Rn, for any n ≥ 3. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the quasihyperbolic metrics are quasiinvariant under a quasisymmetric mapping between two suitable metric spaces. Meanwhile, we also show that quasi-invariance of the quasihyperbolic metrics implies that the corresponding map is quasiconformal. At the end of this paper, as an application of these theorems, we prove that the composition of two quasisymmetric mappings in metric spaces is a quasiconformal mapping. [ABSTRACT FROM AUTHOR]

We obtain Dini conditions that guarantee that an asymptotically conformal quasisphere is rectifiable. In particular, we show that for any ε > 0 integrability of (ess sup1-t<|x|<1+t Kf (x)-1)2-εdt/t implies that the image of the unit sphere under a global quasiconformal homeomorphism f is rectifiable. We also establish estimates for the weak quasisymmetry constant of a global K-quasiconformal map in neighborhoods with maximal dilatation close to 1. [ABSTRACT FROM AUTHOR]

We show that a locally Ahlfors $2$-regular and locally linearly locally contractible metric surface is locally quasisymmetrically equivalent to the disk. We also discuss an application of this result to the problem of characterizing surfaces embedded in some Euclidean spaces that are locally bi-Lipschitz equivalent to a ball in the plane. [ABSTRACT FROM AUTHOR]