Shrinking targets versus recurrence: A survey on the Hausdorff dimension theory

@misc{hu2026shrinking,
  title = {{Shrinking targets versus recurrence: A survey on the Hausdorff dimension theory}},
  author = {Zhang-nan Hu and Jun-jie Huang and Bing Li},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.19546},
  eprint = {2610.19546},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.19546},
}

TY  - UNPB
TI  - Shrinking targets versus recurrence: A survey on the Hausdorff dimension theory
AU  - Zhang-nan Hu
AU  - Jun-jie Huang
AU  - Bing Li
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.19546
UR  - https://arxiv.org/abs/2610.19546
AB  - This survey reviews results on the Hausdorff dimension of shrinking target sets and recurrence sets in dynamical systems. In their basic formulations, shrinking target sets concern orbits approaching a prescribed centre, whereas recurrence sets concern orbits returning close to their initial points. A central theme is the comparison of the Hausdorff dimensions of these two classes of limsup sets. The dynamical systems considered include expanding conformal systems, irrational rotations, affine iterated function systems, and matrix transformations. Related results on Hausdorff measure are also included when they provide natural refinements of the dimension theory. Finally, we review results on badly approximating orbits and non-recurrent sets, with particular attention to winning properties and intersections with self-conformal and self-affine sets.
ER  -