## Intermediate dimensions of infinitely generated attractors

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Amlan Banaji and Jonathan M. Fraser
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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 $\theta \in [0,1]$ 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.

## References

- Amlan Banaji,
*Generalised intermediate dimensions*. Preprint, arXiv:2011.08613v1, 2020. - Amlan Banaji and Jonathan M. Fraser,
*Assouad type dimensions of infinitely generated self-conformal sets*. Preprint, arXiv:2207.11611, 2022. - Amlan Banaji and István Kolossváry,
*Intermediate dimensions of Bedford-Mcmullen carpets with applications to Lipschitz equivalence*. Preprint, arXiv:2111.05625v1, 2021. - Amlan Banaji and Alex Rutar,
*Attainable forms of intermediate dimensions*, Ann. Fenn. Math.**47**(2022), no. 2, 939–960 (English, with English and Finnish summaries). MR**4448745**, DOI 10.54330/afm.120529 - Stuart A. Burrell,
*Dimensions of fractional Brownian images*, J. Theoret. Probab. (to appear), available at 2002.03659v3. Preprint, arXiv:2002.03659v3 2021. - Stuart A. Burrell, Kenneth J. Falconer, and Jonathan M. Fraser,
*Projection theorems for intermediate dimensions*, J. Fractal Geom.**8**(2021), no. 2, 95–116. MR**4261662**, DOI 10.4171/jfg/99 - Stuart A. Burrell, Kenneth J. Falconer, and Jonathan M. Fraser,
*The fractal structure of elliptical polynomial spirals*, Monatsh. Math. (to appear). arXiv:2008.08539, 2020. - Vasileios Chousionis, Dmitriy Leykekhman, and Mariusz Urbański,
*On the dimension spectrum of infinite subsystems of continued fractions*, Trans. Amer. Math. Soc.**373**(2020), no. 2, 1009–1042. MR**4068257**, DOI 10.1090/tran/7984 - Jens Peter Reus Christensen,
*On sets of Haar measure zero in abelian Polish groups*, Israel J. Math.**13**(1972), 255–260 (1973). MR**326293**, DOI 10.1007/BF02762799 - Chih-Yung Chu and Sze-Man Ngai,
*Dimensions in infinite iterated function systems consisting of bi-Lipschitz mappings*, Dyn. Syst.**35**(2020), no. 4, 549–583. MR**4177878**, DOI 10.1080/14689367.2020.1734538 - Lara Daw and George Kerchev,
*Fractal dimensions of the Rosenblatt process*. Preprint, arXiv:2103.04714 2021. - Márton Elekes and Donát Nagy,
*Haar null and Haar meager sets: a survey and new results*, Bull. Lond. Math. Soc.**52**(2020), no. 4, 561–619. MR**4171390**, DOI 10.1112/blms.12340 - K. J. Falconer,
*The Hausdorff dimension of self-affine fractals*, Math. Proc. Cambridge Philos. Soc.**103**(1988), no. 2, 339–350. MR**923687**, DOI 10.1017/S0305004100064926 - Kenneth Falconer,
*Fractal geometry*, 3rd ed., John Wiley & Sons, Ltd., Chichester, 2014. Mathematical foundations and applications. MR**3236784** - Kenneth J. Falconer,
*Intermediate Dimensions: A Survey*, 2021, pp. 469–494. In: Thermodynamic Formalism (eds. M. Pollicott and S. Vaienti). Springer Lecture Notes in Mathematics, vol 2290., DOI 10.1007/978-3-030-74863-0_{1}4 - Kenneth J. Falconer,
*Intermediate dimension of images of sequences under fractional Brownian motion*, Statist. Probab. Lett.**182**(2022), Paper No. 109300, 6. MR**4340821**, DOI 10.1016/j.spl.2021.109300 - Kenneth Falconer, Jonathan Fraser, and Xiong Jin,
*Sixty years of fractal projections*, Fractal geometry and stochastics V, Progr. Probab., vol. 70, Birkhäuser/Springer, Cham, 2015, pp. 3–25. MR**3558147**, DOI 10.1007/978-3-319-18660-3_{1} - Kenneth J. Falconer, Jonathan M. Fraser, and Tom Kempton,
*Intermediate dimensions*, Math. Z.**296**(2020), no. 1-2, 813–830. MR**4140764**, DOI 10.1007/s00209-019-02452-0 - Kenneth J. Falconer,
*A Capacity Approach to Box and Packing Dimensions of Projections and Other Images*, 2020, pp. 1–19. - Richard S. Falk and Roger D. Nussbaum,
*A new approach to numerical computation of Hausdorff dimension of iterated function systems: applications to complex continued fractions*, Integral Equations Operator Theory**90**(2018), no. 5, Paper No. 61, 46. MR**3851775**, DOI 10.1007/s00020-018-2485-z - Jonathan M. Fraser,
*Assouad dimension and fractal geometry*, Cambridge Tracts in Mathematics, vol. 222, Cambridge University Press, Cambridge, 2021. MR**4411274**, DOI 10.1017/9781108778459 - Jonathan M. Fraser,
*Interpolating between dimensions*, Fractal geometry and stochastics VI, Progr. Probab., vol. 76, Birkhäuser/Springer, Cham, [2021] ©2021, pp. 3–24. MR**4237247**, DOI 10.1007/978-3-030-59649-1_{1} - Jonathan M. Fraser,
*On Hölder solutions to the spiral winding problem*, Nonlinearity**34**(2021), no. 5, 3251–3270. MR**4260794**, DOI 10.1088/1361-6544/abe75e - Jonathan M. Fraser and Han Yu,
*New dimension spectra: finer information on scaling and homogeneity*, Adv. Math.**329**(2018), 273–328. MR**3783415**, DOI 10.1016/j.aim.2017.12.019 - R. J. Gardner and R. D. Mauldin,
*On the Hausdorff dimension of a set of complex continued fractions*, Illinois J. Math.**27**(1983), no. 2, 334–345. MR**694647**, DOI 10.1215/ijm/1256046498 - Siegfried Graf, R. Daniel Mauldin, and S. C. Williams,
*The exact Hausdorff dimension in random recursive constructions*, Mem. Amer. Math. Soc.**71**(1988), no. 381, x+121. MR**920961**, DOI 10.1090/memo/0381 - PawełHanus and Mariusz Urbański,
*Complex continued fractions with restricted entries*, Electron. J. Differential Equations (1998), No. 27, 9. MR**1649340** - Stefan-M. Heinemann and Mariusz Urbański,
*Hausdorff dimension estimates for infinite conformal IFSs*, Nonlinearity**15**(2002), no. 3, 727–734. MR**1901102**, DOI 10.1088/0951-7715/15/3/312 - Michael Hochman and Pablo Shmerkin,
*Local entropy averages and projections of fractal measures*, Ann. of Math. (2)**175**(2012), no. 3, 1001–1059. MR**2912701**, DOI 10.4007/annals.2012.175.3.1 - John E. Hutchinson,
*Fractals and self-similarity*, Indiana Univ. Math. J.**30**(1981), no. 5, 713–747. MR**625600**, DOI 10.1512/iumj.1981.30.30055 - Daniel Ingebretson,
*Quantitative distortion and the Hausdorff dimension of continued fractions*. Preprint, arXiv:2002.10232 2020. - Antti Käenmäki and Henry W. J. Reeve,
*Multifractal analysis of Birkhoff averages for typical infinitely generated self-affine sets*, J. Fractal Geom.**1**(2014), no. 1, 83–152. MR**3166207**, DOI 10.4171/JFG/3 - Jean-Pierre Kahane,
*Some random series of functions*, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 5, Cambridge University Press, Cambridge, 1985. MR**833073** - Marc Kesseböhmer and Sanguo Zhu,
*Dimension sets for infinite IFSs: the Texan conjecture*, J. Number Theory**116**(2006), no. 1, 230–246. MR**2197868**, DOI 10.1016/j.jnt.2005.04.002 - R. Daniel Mauldin,
*Infinite iterated function systems: theory and applications*, Fractal geometry and stochastics (Finsterbergen, 1994) Progr. Probab., vol. 37, Birkhäuser, Basel, 1995, pp. 91–110. MR**1391972**, DOI 10.1007/978-3-0348-7755-8_{5} - R. Daniel Mauldin and Mariusz Urbański,
*Conformal iterated function systems with applications to the geometry of continued fractions*, Trans. Amer. Math. Soc.**351**(1999), no. 12, 4995–5025. MR**1487636**, DOI 10.1090/S0002-9947-99-02268-0 - R. Daniel Mauldin and Mariusz Urbański,
*Dimensions and measures in infinite iterated function systems*, Proc. London Math. Soc. (3)**73**(1996), no. 1, 105–154. MR**1387085**, DOI 10.1112/plms/s3-73.1.105 - Sze-Man Ngai and Ji-Xi Tong,
*Infinite iterated function systems with overlaps*, Ergodic Theory Dynam. Systems**36**(2016), no. 3, 890–907. MR**3480349**, DOI 10.1017/etds.2014.86 - William Ott and James A. Yorke,
*Prevalence*, Bull. Amer. Math. Soc. (N.S.)**42**(2005), no. 3, 263–290. MR**2149086**, DOI 10.1090/S0273-0979-05-01060-8 - Amit Priyadarshi,
*Lower bound on the Hausdorff dimension of a set of complex continued fractions*, J. Math. Anal. Appl.**449**(2017), no. 1, 91–95. MR**3595193**, DOI 10.1016/j.jmaa.2016.12.009 - Pablo Shmerkin,
*Projections of self-similar and related fractals: a survey of recent developments*, Fractal geometry and stochastics V, Progr. Probab., vol. 70, Birkhäuser/Springer, Cham, 2015, pp. 53–74. MR**3558150**, DOI 10.1007/978-3-319-18660-3_{4} - Justin T. Tan,
*On the intermediate dimensions of concentric spheres and related sets*. Preprint, arXiv:2008.10564 2020.

## 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