Intermediate dimensions of infinitely generated attractors

Amlan Banaji, Jonathan Fraser*

*Corresponding author for this work

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We study the dimension theory of limit sets of iterated function systems consisting of a countably infinite number of contractions. Our primary focus is on the intermediate dimensions: a family of dimensions depending on a parameter θ ε [0; 1] which interpolate between the Hausdorff and box dimensions. Our main results are in the case when all the contractions are conformal. Under a natural separation condition we prove that the intermediate dimensions of the limit set are the maximum of the Hausdorff dimension of the limit set and the intermediate dimensions of the set of fixed points of the contractions. This builds on work of Mauldin and Urbanski concerning the Hausdorff and upper box dimension. We give several (often counter-intuitive) applications of our work to dimensions of projections, fractional Brownian images, and general Hölder images. These applications apply to well-studied examples such as sets of numbers which have real or complex continued fraction expansions with restricted entries. 
We also obtain several results without assuming conformality or any separation conditions. We prove general upper bounds for the Hausdorff, box and intermediate dimensions of infinitely generated attractors in terms of a topological pressure function. We also show that the limit set of a ‘generic’ infinite iterated function system has box and intermediate dimensions equal to the ambient spatial dimension, where ‘generic’ can refer to any one of (i) full measure; (ii) prevalent; or (iii) comeagre.
Original languageEnglish
Pages (from-to)2449-2479
Number of pages31
JournalTransactions of the American Mathematical Society
Issue number4
Early online date24 Jan 2023
Publication statusPublished - 1 Apr 2023


  • Infinite iterated function system
  • Conformal iterated function system
  • Intermediate dimensions
  • Hausdorff dimension
  • Box dimension
  • Topological pressure function
  • Continued fractions


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