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We give study the Lipschitz continuity of Mobius transformations of a punctured disk onto another punctured disk with respect to the distance ratio metric.
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We study expansion/contraction properties of some common classes of mappings of the Euclidean space ${mathbb R}^n, nge 2,,$ with respect to the distance ratio metric. The first main case is the behavior of Mobius transformations of the unit ball in ${mathbb R}^n$ onto itself. In the second main case we study the polynomials of the unit disk onto a subdomain of the complex plane. In both cases sharp Lipschitz constants are obtained.
The triangular ratio metric is studied in a domain $Gsubsetneqmathbb{R}^n$, $ngeq2$. Several sharp bounds are proven for this metric, especially, in the case where the domain is the unit disk of the complex plane. The results are applied to study the Holder continuity of quasiconformal mappings.
Suppose that $E$ and $E$ denote real Banach spaces with dimension at least $2$ and that $Dsubset E$ and $Dsubset E$ are domains. In this paper, we establish, in terms of the $j_D$ metric, a necessary and sufficient condition for the homeomorphism $f: E to E$ to be FQC. Moreover, we give, in terms of the $j_D$ metric, a sufficient condition for the homeomorphism $f: Dto D$ to be FQC. On the other hand, we show that this condition is not necessary.
We prove the differentiability of Lipschitz maps X-->V, where X is a complete metric measure space satisfying a doubling condition and a Poincare inequality, and V is a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure spaces, in terms of directional derivatives in the direction of tangent vectors to suitable rectifiable curves.
In this paper we provide several emph{metric universality} results. We exhibit for certain classes $cC$ of metric spaces, families of metric spaces $(M_i, d_i)_{iin I}$ which have the property that a metric space $(X,d_X)$ in $cC$ is coarsely, resp. Lipschitzly, universal for all spaces in $cC$ if the collection of spaces $(M_i,d_i)_{iin I}$ equi-coarsely, respectively equi-Lipschitzly, embeds into $(X,d_X)$. Such families are built as certain Schreier-type metric subsets of $co$. We deduce a metric analog to Bourgains theorem, which generalized Szlenks theorem, and prove that a space which is coarsely universal for all separable reflexive asymptotic-$c_0$ Banach spaces is coarsely universal for all separable metric spaces. One of our coarse universality results is valid under Martins Axiom and the negation of the Continuum Hypothesis. We discuss the strength of the universality statements that can be obtained without these additional set theoretic assumptions. In the second part of the paper, we study universality properties of Kaltons interlacing graphs. In particular, we prove that every finite metric space embeds almost isometrically in some interlacing graph of large enough diameter.
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