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PreprintarXiv licencearXiv

Intersections of quantitative recurrence and badly approximable sets for expanding Markov maps

  • Lingmin Liao
  • Na Yuan
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Abstract

We study a class of expanding Markov maps on the unit interval, which includes sofic β-transformations, Gauss maps, tent maps and cookie-cutter maps. We investigate the quantitative recurrence property for points that are dynamically badly approximable, namely, points whose orbits eventually remain a positive distance from a given point. We show that the Hausdorff dimension of the intersection of a recurrence set and a badly approximable set is determined by the same pressure function as that of the corresponding recurrence set. We further obtain the Hausdorff dimensions of two related intersections: recurrence sets with non-recurrent sets, and shrinking target sets with badly approximable sets.

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Cite

BibTeXRIS
BibTeX
@misc{liao2026intersections,
  title = {{Intersections of quantitative recurrence and badly approximable sets for expanding Markov maps}},
  author = {Lingmin Liao and Na Yuan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12258},
  eprint = {2610.12258},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12258},
}
RIS
TY  - UNPB
TI  - Intersections of quantitative recurrence and badly approximable sets for expanding Markov maps
AU  - Lingmin Liao
AU  - Na Yuan
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12258
UR  - https://arxiv.org/abs/2610.12258
AB  - We study a class of expanding Markov maps on the unit interval, which includes sofic β-transformations, Gauss maps, tent maps and cookie-cutter maps. We investigate the quantitative recurrence property for points that are dynamically badly approximable, namely, points whose orbits eventually remain a positive distance from a given point. We show that the Hausdorff dimension of the intersection of a recurrence set and a badly approximable set is determined by the same pressure function as that of the corresponding recurrence set. We further obtain the Hausdorff dimensions of two related intersections: recurrence sets with non-recurrent sets, and shrinking target sets with badly approximable sets.
ER  -