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4 results

  • The Assouad dimension of self-affine measures on sponges

  • Part of
  • JONATHAN M. FRASER, ISTVÁN KOLOSSVÁRY
  • Journal:
    Ergodic Theory and Dynamical Systems / Volume 43 / Issue 9 / September 2023
    Published online by Cambridge University Press:
    08 September 2022, pp. 2974-2996
    Print publication:
    September 2023
    • Article
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  • We derive upper and lower bounds for the Assouad and lower dimensions of self-affine measures in $\mathbb {R}^d$ generated by diagonal matrices and satisfying suitable separation conditions. The upper and lower bounds always coincide for $d=2,3$, yielding precise explicit formulae for those dimensions. Moreover, there are easy-to-check conditions guaranteeing that the bounds coincide for $d \geqslant 4$. An interesting consequence of our results is that there can be a ‘dimension gap’ for such self-affine constructions, even in the plane. That is, we show that for some self-affine carpets of ‘Barański type’ the Assouad dimension of all associated self-affine measures strictly exceeds the Assouad dimension of the carpet by some fixed $\delta>0$ depending only on the carpet. We also provide examples of self-affine carpets of ‘Barański type’ where there is no dimension gap and in fact the Assouad dimension of the carpet is equal to the Assouad dimension of a carefully chosen self-affine measure.

  • 4 - Dimensions of Measures

  • 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 56-63

  • Summary

    A key aspect of dimension theory is how dimensions of sets relate to dimensions of measures. To this end, all the familiar notions of dimension for sets have natural counterparts in the context of measures. The Assouad dimension and its many variants are no different and in this chapter we explore the dimensions and spectra of measures.

Assouad Dimension and Fractal Geometry
  • Jonathan M. Fraser
  • Published online:
    13 October 2020
    Print publication:
    29 October 2020
  • The Assouad dimension is a notion of dimension in fractal geometry that has been the subject of much interest in recent years. This book, written by a world expert on the topic, is the first thorough account of the Assouad dimension and its many variants and applications in fractal geometry and beyond. It places the theory of the Assouad dimension in context among up-to-date treatments of many key advances in fractal geometry, while also emphasising its diverse connections with areas of mathematics including number theory, dynamical systems, harmonic analysis, and probability theory. A final chapter detailing open problems and future directions for research brings readers to the cutting edge of this exciting field. This book will be an indispensable part of the modern fractal geometer's library and a valuable resource for pure mathematicians interested in the beauty and many applications of the Assouad dimension.