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Abstract
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 counterintuitive) applications of our work to dimensions of projections, fractional Brownian images, and general Hölder images. These applications apply to wellstudied 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.
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 language  English 

Pages (fromto)  24492479 
Number of pages  31 
Journal  Transactions of the American Mathematical Society 
Volume  376 
Issue number  4 
Early online date  24 Jan 2023 
DOIs  
Publication status  Published  1 Apr 2023 
Keywords
 Infinite iterated function system
 Conformal iterated function system
 Intermediate dimensions
 Hausdorff dimension
 Box dimension
 Topological pressure function
 Continued fractions
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Dive into the research topics of 'Intermediate dimensions of infinitely generated attractors'. Together they form a unique fingerprint.Projects
 3 Finished


New perspectives in the dimension: New perspectives in the dimension theory of fractals
1/09/19 → 31/01/23
Project: Standard

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis
1/02/18 → 11/06/21
Project: Standard