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Intermediate dimensions of infinitely generated attractors (2104.15133v4)

Published 30 Apr 2021 in math.DS, math.CA, and math.MG

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 Urba\'nski 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 H\"older 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 mean either 'full measure' or 'comeagre.'

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References (41)
  1. Amlan Banaji “Generalised intermediate dimensions” In Monatsh. Math. 202, 2023, pp. 465–506
  2. Amlan Banaji and Jonathan M. Fraser “Assouad type dimensions of infinitely generated self-conformal sets” In Nonlinearity 37.4, 2024, pp. 045004
  3. “Intermediate dimensions of Bedford-McMullen carpets with applications to Lipschitz equivalence” Preprint arXiv:2111.05625v1 [math.DS]
  4. “Attainable forms of intermediate dimensions” In Ann. Fenn. Math. 47.2, 2022, pp. 939–960
  5. Stuart A. Burrell “Dimensions of fractional Brownian images” In J. Theoret. Probab. 35, 2022, pp. 2217–2238
  6. Stuart A. Burrell, Kenneth J. Falconer and Jonathan M. Fraser “Projection theorems for intermediate dimensions” In J. Fractal Geom. 8, 2021, pp. 95–116
  7. Stuart A. Burrell, Kenneth J. Falconer and Jonathan M. Fraser “The fractal structure of elliptical polynomial spirals” In Monatsh. Math. 199, 2022, pp. 1–22
  8. Vasilis Chousionis, Dmitriy Leykekhman and Mariusz Urbański “On the dimension spectrum of infinite subsystems of continued fractions” In Trans. Amer. Math. Soc. 373.2, 2020, pp. 1009–1042 DOI: 10.1090/tran/7984
  9. “Dimensions in infinite iterated function systems consisting of bi-Lipschitz mappings” In Dyn. Syst. 35, 2020, pp. 549–583
  10. “Fractal dimensions of the Rosenblatt process” In Stochastic Process. Appl. 161, 2023, pp. 544–571
  11. “Haar null and Haar meager sets: a survey and new results” In Bull. London Math. Soc. 52, 2020, pp. 561–619
  12. Kenneth J. Falconer “Fractal Geometry: Mathematical Foundations and Applications” Wiley, 2014
  13. Kenneth J. Falconer “Intermediate dimension of images of sequences under fractional Brownian motion” In Statist. Probab. Lett. 182, 2022 DOI: https://doi.org/10.1016/j.spl.2021.109300
  14. Kenneth J. Falconer “Intermediate Dimensions: A Survey” In Thermodynamic Formalism (eds. M. Pollicott and S. Vaienti) Springer Lecture Notes in Mathematics, vol 2290, 2021, pp. 469–494
  15. Kenneth J. Falconer “The Hausdorff dimension of self-affine fractals” In Math. Proc. Cambridge Philos. Soc. 103, 1988, pp. 339–350
  16. Kenneth J. Falconer, Jonathan M. Fraser and Xiong Jin “Sixty years of fractal projections” In Fractal Geometry and Stochastics V (eds. C. Bandt, K. J. Falconer and M. Zähle) 70 Birkhäuser, Progr. Probab., 2015, pp. 3–25 DOI: 10.1007/978-3-319-18660-3\_1
  17. Kenneth J. Falconer, Jonathan M. Fraser and Tom Kempton “Intermediate dimensions” In Math. Z. 296, 2020, pp. 813–830 DOI: https://doi.org/10.1007/s00209-019-02452-0
  18. Kenneth John Falconer “A Capacity Approach to Box and Packing Dimensions of Projections and Other Images” In Analysis, Probability and Mathematical Physics on Fractals (eds. P. A. Ruiz, J. P. Chen, L. G. Rogers and R. S. Strichartz), Fractals and Dynamics in Mathematics, Science and the Arts: Theory and Applications World Scientific, 2020, pp. 1–19 DOI: 10.1142/11696
  19. “A New Approach to Numerical Computation of Hausdorff Dimension of Iterated Function Systems: Applications to Complex Continued Fractions” In Integral Equations Operator Theory 90, 2018 DOI: 10.1007/s00020-018-2485-z
  20. Jonathan M. Fraser “Assouad Dimension and Fractal Geometry” Cambridge University Press, Tracts in Mathematics Series, 222, 2020
  21. Jonathan M. Fraser “Interpolating Between Dimensions” In Fractal Geometry and Stochastics VI (eds. U. Freiberg, B. Hambly, M. Hinz and S. Winter) Birkhäuser, Progr. Probab., vol 76, 2021
  22. Jonathan M. Fraser “On Hölder solutions to the spiral winding problem” In Nonlinearity 34, 2021, pp. 3251–3270
  23. Jonathan M. Fraser and Han Yu “New dimension spectra: Finer information on scaling and homogeneity” In Adv. Math. 329, 2018, pp. 273–328 DOI: 10.1016/j.aim.2017.12.019
  24. “On the Hausdorff dimension of a set of complex continued fractions” In Illinois J. Math. 27.2 Duke University Press, 1983, pp. 334–345 DOI: 10.1215/ijm/1256046498
  25. Siegfried Graf, R.Daniel Mauldin and Stanley C. Williams “The exact Hausdorff dimension in random recursive constructions” In Mem. Amer. Math. Soc. 71, 1988
  26. “Complex continued fractions with restricted entries” In Electron. J. Differential Equations 27, 1998, pp. 1–9
  27. “Hausdorff dimension estimates for infinite conformal IFSs” In Nonlinearity 15, 2002, pp. 727–734
  28. “Local entropy averages and projections of fractal measures” In Ann. of Math. (2) 175.3, 2012, pp. 1001–1059 DOI: 10.4007/annals.2012.175.3.1
  29. John E. Hutchinson “Fractals and self-similarity” In Indiana Univ. Math. J. 30, 1981, pp. 713–747
  30. Daniel Ingebretson “Quantitative distortion and the Hausdorff dimension of continued fractions” Preprint arXiv:2002.10232 [math.NT]
  31. Antti Käenmäki and Henry W.J. Reeve “Multifractal analysis of Birkhoff averages for typical infinitely generated self-affine sets” In J. Fractal Geom. 1, 2014, pp. 83–152
  32. J.P. Kahane “Some Random Series of Functions” Cambridge University Press, 1985
  33. “Dimension sets for infinite IFSs: the Texan Conjecture” In J. Number Theory 116.1, 2006, pp. 230–246 DOI: https://doi.org/10.1016/j.jnt.2005.04.002
  34. R.Daniel Mauldin “Infinite Iterated Function Systems: Theory and Applications” In Fractal Geometry and Stochastics (eds. C. Bandt, S. Graf and M. Zähle) 37 Birkhäuser, Progr. Probab., 1995
  35. “Conformal iterated function systems with applications to the geometry of continued fractions” In Trans. Amer. Math. Soc. 351, 1999, pp. 4995–5025
  36. “Dimensions and Measures in Infinite Iterated Function Systems” In Proc. Lond. Math. Soc. s3-73.1, 1996, pp. 105–154 DOI: 10.1112/plms/s3-73.1.105
  37. “Infinite iterated function systems with overlaps” In Ergodic Theory Dynam. Systems 36.3 Cambridge University Press, 2016, pp. 890–907 DOI: 10.1017/etds.2014.86
  38. William Ott and James A. Yorke “Prevalence” In Bull. Amer. Math. Soc. (N.S.) 42, 2005, pp. 263–290
  39. Amit Priyadarshi “Lower bound on the Hausdorff dimension of a set of Complex Continued Fractions” In J. Math. Anal. Appl. 449, 2016 DOI: 10.1016/j.jmaa.2016.12.009
  40. Pablo Shmerkin “Projections of self-similar and related fractals: a survey of recent developments” In Fractal Geometry and Stochastics V (eds. C. Bandt, K. J. Falconer and M. Zähle) 70 Birkhäuser, Progr. Probab., 2015, pp. 53–74 DOI: 10.1007/978-3-319-18660-3\_4
  41. Justin T. Tan “On the intermediate dimensions of concentric spheres and related sets” Preprint arXiv:2008.10564 [math.MG]
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