Intermediate dimensions of infinitely generated attractors
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- by Amlan Banaji and Jonathan M. Fraser HTML | PDF
- Trans. Amer. Math. Soc. 376 (2023), 2449-2479 Request permission
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 $\theta \in [0,1]$ 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 Urbański 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.
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Additional Information
- Amlan Banaji
- Affiliation: School of Mathematics and Statistics, University of St Andrews, Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, United Kingdom
- ORCID: 0000-0002-3727-0894
- Email: afb8@st-andrews.ac.uk
- Jonathan M. Fraser
- Affiliation: School of Mathematics and Statistics, University of St Andrews, Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, United Kingdom
- MR Author ID: 946983
- ORCID: 0000-0002-8066-9120
- Email: jmf32@st-andrews.ac.uk
- Received by editor(s): May 6, 2021
- Received by editor(s) in revised form: February 22, 2022, and April 25, 2022
- Published electronically: January 24, 2023
- Additional Notes: Both authors were financially supported by a Leverhulme Trust Research Project Grant (RPG-2019-034), and the second author was also supported by an EPSRC Standard Grant (EP/R015104/1).
- © Copyright 2023 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 376 (2023), 2449-2479
- MSC (2020): Primary 28A80; Secondary 37B10, 11K50
- DOI: https://doi.org/10.1090/tran/8766
- MathSciNet review: 4557871