Hostname: page-component-76c49bb84f-c7tcl Total loading time: 0 Render date: 2025-07-05T06:51:59.112Z Has data issue: false hasContentIssue false
  • English
  • Français

DIMENSIONAL APPROXIMATION OF AN INHOMOGENEOUS ATTRACTOR WITHOUT ANY SEPARATION CONDITION

Part of: Classical measure theory

Published online by Cambridge University Press:  23 June 2025

MANUJ VERMA Open the ORCID record for MANUJ VERMA [Opens in a new window]
MANUJ VERMA*
Affiliation:
Department of Stochastics, Institute of Mathematics, https://ror.org/02w42ss30Budapest University of Technology and Economics, Műegyetem rpk. 1-3, Budapest H-1111, Hungary Department of Mathematics, https://ror.org/049tgcd06Indian Institute of Technology Delhi, New Delhi 110016, India
*
e-mail: mathmanuj@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that the Hausdorff dimension of the attractor of an inhomogeneous self-similar iterated function system (or self-similar IFS) can be well approximated by the Hausdorff dimension of the attractor of another inhomogeneous self-similar IFS satisfying the strong separation condition. We also determine a formula for the Hausdorff dimension of the algebraic product and sum of the inhomogeneous attractor.

Keywords

Hausdorff dimensioninhomogeneous iterated function systemstrong separation conditionsimilarity dimension

MSC classification

Primary: 28A80: Fractals
Secondary: 28A78: Hausdorff and packing measures

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 13
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

The Author acknowledges support from the grant NKFI KKP144059 ‘Fractal geometry and applications’.

References

Baker, S., Fraser, J. M. and Máthé, A., ‘Inhomogeneous self-similar sets with overlaps’, Ergodic Theory Dynam. Systems 39(1) (2019), 1–18.10.1017/etds.2017.13CrossRefGoogle Scholar
Bárány, B., ‘On some non-linear projections of self-similar sets in ${\mathbb{R}}^3$ ’, Fund. Math. 237 (2017), 83–100.10.4064/fm90-4-2016CrossRefGoogle Scholar
Bárány, B., Käenmäki, A. and Nissinen, P., ‘Covering number on inhomogeneous graph-directed self-similar sets’, Preprint, 2023, arXiv:2307.16263.Google Scholar
Barnsley, M. F. and Demko, S., ‘Iterated function systems and the global construction of fractals’, Proc. Roy. Soc. Lond. Sect. A 399 (1985), 243–275.Google Scholar
Chandra, S. and Abbas, S., ‘On fractal dimensions of fractal functions using function spaces’, Bull. Aust. Math. Soc. 106 (2022), 470–480.10.1017/S0004972722000685CrossRefGoogle Scholar
Dubey, S. and Verma, S., ‘Fractal dimension for a class of inhomogeneous graph-directed attractors’, Discrete and Continuous Dynamical Systems - Series S (2024), doi:10.3934/dcdss.2024158.CrossRefGoogle Scholar
Dubey, S. and Verma, S., ‘Inhomogeneous graph-directed attractors and fractal measures’, J. Anal. 32 (2024), 157–170.10.1007/s41478-023-00614-2CrossRefGoogle Scholar
Falconer, K. J., ‘The Hausdorff dimension of some fractals and attractors of overlapping construction’, J. Stat. Phys. 47 (1987), 123–132.10.1007/BF01009037CrossRefGoogle Scholar
Falconer, K. J., Techniques in Fractal Geometry (Wiley, Chichester, 1997).Google Scholar
Falconer, K. J., Fractal Geometry: Mathematical Foundations and Applications (Wiley, New York, 1999).Google Scholar
Farkas, Á., ‘Projections of self-similar sets with no separation condition’, Israel J. Math. 214(1) (2016), 67–107.10.1007/s11856-016-1345-2CrossRefGoogle Scholar
Farkas, Á., ‘Dimension approximation of attractors of graph directed IFSs by self-similar sets’, Math. Proc. Cambridge Philos. Soc. 167(1) (2019), 193–207.10.1017/S0305004118000282CrossRefGoogle Scholar
Fraser, J. M., ‘Inhomogeneous self-similar sets and box dimensions’, Studia Math. 213 (2012), 133–156.10.4064/sm213-2-2CrossRefGoogle Scholar
Fraser, J. M., Assouad Dimension and Fractal Geometry, Cambridge Tracts in Pure Mathematics, 222 (Cambridge University Press, Cambridge, 2020).10.1017/9781108778459CrossRefGoogle Scholar
Hutchinson, J. E., ‘Fractals and self-similarity’, Indiana Univ. Math. J. 30 (1981), 713–747.10.1512/iumj.1981.30.30055CrossRefGoogle Scholar
Jha, S. and Verma, S., ‘Dimensional analysis of $\alpha$ -fractal functions’, Results Math. 76(4) (2021), 1–24.10.1007/s00025-021-01495-2CrossRefGoogle Scholar
Mauldin, R. D. and Urbański, M., ‘Dimensions and measures in infinite iterated function systems’, Proc. Lond. Math. Soc. (3) 73 (1996), 105–154.10.1112/plms/s3-73.1.105CrossRefGoogle Scholar
Mauldin, R. D. and Urbański, M., ‘Conformal iterated function systems with applications to the geometry of continued fractions’, Trans. Amer. Math. Soc. 351(12) (1999), 4995–5025.10.1090/S0002-9947-99-02268-0CrossRefGoogle Scholar
Mauldin, R. D. and Williams, S. C., ‘Hausdorff dimension in graph directed construction’, Trans. Amer. Math. Soc. 309(2) (1988), 811–829.CrossRefGoogle Scholar
Nussbaum, R. D., Priyadarshi, A. and Lunel, S. V., ‘Positive operators and Hausdorff dimension of invariant sets’, Trans. Amer. Math. Soc. 364(2) (2012), 1029–1066.10.1090/S0002-9947-2011-05484-XCrossRefGoogle Scholar
Olsen, L. and Snigireva, N., ‘ ${L}^q$ spectra and Rényi dimensions of in-homogeneous self-similar measures’, Nonlinearity 20 (2007), 151–175.10.1088/0951-7715/20/1/010CrossRefGoogle Scholar
Orponen, T., ‘On the distance sets of self-similar sets’, Nonlinearity 25 (2012), 1919–1929.10.1088/0951-7715/25/6/1919CrossRefGoogle Scholar
Peres, Y. and Shmerkin, P., ‘Resonance between Cantor sets’, Ergodic Theory Dynam. Systems 29 (2009), 201–221.CrossRefGoogle Scholar
Priyadarshi, A., ‘Lower bound on the Hausdorff dimension of a set of complex continued fractions’. J. Math. Anal. Appl. 449(1) (2017), 91–95.10.1016/j.jmaa.2016.12.009CrossRefGoogle Scholar
Verma, S., ‘Hausdorff dimension and infinitesimal similitudes on complete metric spaces’, Preprint, 2021, arXiv:2101.07520.Google Scholar