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PreprintCC BY-NC-SA 4.0arXiv

Dimension properties of a family of random self-similar iterated function systems

  • Vilma Orgoványi
  • Károly Simon
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Abstract

In this note we present a dimension formula for certain randomized self-similar sets. The self-similar sets we are considering are d-dimensional generalizations of rational projections of inhomogeneous sponges: they have overlaps, but the overlapping structure is in some sense well-organized. The randomization happens according to a labelled Galton-Watson tree corresponding to the cylinders of the IFS, for example as in the case of the Mandelbrot percolation set. The dimension formula is given in terms of a pressure function corresponding to the expectation matrices, which describes the overlapping structure and the randomness of the system.

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Cite

BibTeXRIS
BibTeX
@misc{orgovanyi2026dimension,
  title = {{Dimension properties of a family of random self-similar iterated function systems}},
  author = {Vilma Orgoványi and Károly Simon},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08465},
  eprint = {2610.08465},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08465},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Dimension properties of a family of random self-similar iterated function systems
AU  - Vilma Orgoványi
AU  - Károly Simon
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.08465
UR  - https://arxiv.org/abs/2610.08465
AB  - In this note we present a dimension formula for certain randomized self-similar sets. The self-similar sets we are considering are d-dimensional generalizations of rational projections of inhomogeneous sponges: they have overlaps, but the overlapping structure is in some sense well-organized. The randomization happens according to a labelled Galton-Watson tree corresponding to the cylinders of the IFS, for example as in the case of the Mandelbrot percolation set. The dimension formula is given in terms of a pressure function corresponding to the expectation matrices, which describes the overlapping structure and the randomness of the system.
ER  -