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3 results

  • Teichmüller displacement theorem on Gromov hyperbolic spaces

  • Part of
  • Qingshan Zhou, Saminathan Ponnusamy, Qianghua Luo
  • Journal:
    Proceedings of the Edinburgh Mathematical Society , First View
    Published online by Cambridge University Press:
    06 November 2024, pp. 1-19
    • Article
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  • Given a Gromov hyperbolic domain $G\subsetneq \mathbb{R}^n$ with uniformly perfect Gromov boundary, Zhou and Rasila recently proved that for all quasiconformal homeomorphisms $\psi\colon G\to G$ with identity value on the Gromov boundary, the quasihyperbolic displacement $k_G(x,\psi(x))$ for all $x\in G$ is bounded above. In this paper, we generalize this result and establish Teichmüller displacement theorem for quasi-isometries of Gromov hyperbolic spaces in a quantitative way. As applications, we obtain its connections to bilipschitz extensions of certain Gromov hyperbolic spaces.

  • 13 - Applications in Embedding Theory

  • from Part Three - Applications
  • Jonathan M. Fraser, University of St Andrews, Scotland
  • Book:
    Assouad Dimension and Fractal Geometry
    Published online:
    13 October 2020
    Print publication:
    29 October 2020, pp 197-214

  • Summary

    Even though the Assouad dimension appeared in other contexts, it was the pioneering work of Assouad in the 1970s which served to popularise the notion and establish its deep connections with several areas of mathematics, most notably embedding theory. In this chapter we discuss various problems in embedding theory, all of which can be understood as asking for conditions on a metric space guaranteeing the existence of an injection (the embedding) into Euclidean space.

  • ON THE UNIFORM PERFECTNESS OF THE BOUNDARY OF MULTIPLY CONNECTED WANDERING DOMAINS

  • Part of
  • WALTER BERGWEILER, JIAN-HUA ZHENG
  • Journal:
    Journal of the Australian Mathematical Society / Volume 91 / Issue 3 / December 2011
    Published online by Cambridge University Press:
    04 November 2011, pp. 289-311
    Print publication:
    December 2011
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  • We investigate when the boundary of a multiply connected wandering domain of an entire function is uniformly perfect. We give a general criterion implying that it is not uniformly perfect. This criterion applies in particular to examples of multiply connected wandering domains given by Baker. We also provide examples of infinitely connected wandering domains whose boundary is uniformly perfect.