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4 results

  • Quasisymmetrically rigid self-similar carpets

  • Part of
  • FENG RAO, FAN WEN
  • Journal:
    Ergodic Theory and Dynamical Systems , First View
    Published online by Cambridge University Press:
    22 October 2024, pp. 1-14
    • Article
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  • We define balanced self-similar quasi-round carpets and compare the carpet moduli of some path families relating to such a carpet. Then, using some known results on quasiconformal geometry of carpets, we prove that the group of quasisymmetric self-homeomorphisms of every balanced self-similar quasi-round carpet is finite. Furthermore, we prove that some balanced self-similar carpets in the unit square with strong geometric symmetry are quasisymmetrically rigid by using the quasisymmetry of weak tangents of carpets.

  • 12 - Conformal Dimension

  • from Part Two - Examples
  • 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 190-196

  • Summary

    In this chapter we briefly discuss conformal dimension and highlight some particular areas of interest in connection with the topics covered in this book. The concept of conformal dimension considers how much a given notion of dimension can drop under quasi-symmetric deformation.

  • 5 - Weak Tangents and Microsets

  • from Part One - Theory
  • 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 64-78

  • Summary

    One of the most effective ways to bound the Assouad dimension of a set from below is to use weak tangents; an approach pioneered by Mackay and Tyson. Weak tangents are tools for capturing the extremal local structure of a set. In this chapter we explore this connection. We prove Mackay and Tyson's estimate, and a stronger result of Furstenberg which characterises the Assouad dimension entirely in terms of the Hausdorffdimension of weak tangents.