The fractal structure of elliptical polynomial spirals

Stuart Andrew Burrell*, Kenneth John Falconer, Jonathan Fraser

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We investigate fractal aspects of elliptical polynomial spirals, that is planar spirals with differing polynomial rates of decay in the two axis directions. We give a full dimensional analysis of these spirals, computing explicitly their intermediate, box-counting and Assouad-type dimensions. An exciting feature is that these spirals exhibit two phase transitions within the Assouad spectrum, the first natural class of fractals known to have this property. We go on to use this dimensional information to obtain bounds for the Hölder regularity of maps that can deform one spiral into another, generalising the 'winding problem' of when spirals are bi-Lipschitz equivalent to a line segment. A novel feature is the use of fractional Brownian motion and dimension profiles to bound the Hölder exponents.
Original languageEnglish
Pages (from-to)1-22
JournalMonatshefte für Mathematik
Early online date3 Jul 2022
Publication statusPublished - 1 Sept 2022


  • Elliptical polynomial spiral
  • Generalised hyperbolic spiral
  • Box-counting dimension
  • Assouad dimension
  • Assouad spectrum
  • Intermediate dimensions
  • Hölder exponents
  • Fractional Brownian motion


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