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PreprintarXiv licencearXiv

Markov-driven self-similar measures and non-homogeneous slicing

  • Pablo Shmerkin
  • Shunsuke Usuki
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Abstract

We introduce the concept of Markov-driven self-similar measures, a notion that simultaneously generalizes classical self-similar measures, self-conformal measures, dynamically driven self-similar measures and many random measures. This class is also closed under Cartesian products and linear projections. We initiate the study of this class of measures and investigate the Lq dimension of Markov-driven self-similar measures under certain Diophantine conditions. As an application, we obtain the first Furstenberg slicing type results for intersections of Cantor sets in non-homogeneous settings.

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Cite

BibTeXRIS
BibTeX
@misc{shmerkin2026markovdriven,
  title = {{Markov-driven self-similar measures and non-homogeneous slicing}},
  author = {Pablo Shmerkin and Shunsuke Usuki},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.00962},
  eprint = {2611.00962},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.00962},
}
RIS
TY  - UNPB
TI  - Markov-driven self-similar measures and non-homogeneous slicing
AU  - Pablo Shmerkin
AU  - Shunsuke Usuki
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.00962
UR  - https://arxiv.org/abs/2611.00962
AB  - We introduce the concept of Markov-driven self-similar measures, a notion that simultaneously generalizes classical self-similar measures, self-conformal measures, dynamically driven self-similar measures and many random measures. This class is also closed under Cartesian products and linear projections. We initiate the study of this class of measures and investigate the L^q dimension of Markov-driven self-similar measures under certain Diophantine conditions. As an application, we obtain the first Furstenberg slicing type results for intersections of Cantor sets in non-homogeneous settings.
ER  -