Minimal sets for torus homeomorphisms with an irrational circle factor

@misc{liu2026minimal,
  title = {{Minimal sets for torus homeomorphisms with an irrational circle factor}},
  author = {Xiao-Chuan Liu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23972},
  eprint = {2609.23972},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23972},
  note = {CC0},
}

TY  - UNPB
TI  - Minimal sets for torus homeomorphisms with an irrational circle factor
AU  - Xiao-Chuan Liu
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.23972
UR  - https://arxiv.org/abs/2609.23972
AB  - We study minimal sets of torus homeomorphisms admitting an irrational circle factor whose fibres are thin essential annular continua. For totally irrational pseudo-rotations and homeomorphisms in a nontrivial Dehn-twist class, we prove uniqueness of the minimal set when the fibres are Jordan curves on a residual set of base parameters. The same conclusion holds if the fibre cores are Jordan curves, or if the fibres are locally connected, on a nonmeagre set of parameters. The residual hypothesis cannot be replaced by a full-measure hypothesis, even under area preservation and topological transitivity. For every totally irrational rotation vector, we construct such a map with Jordan-curve fibres almost everywhere and uncountably many pairwise disjoint uniquely ergodic minimal Cantor sets. We also construct examples in every nontrivial Dehn-twist class, with prescribed irrational vertical rotation number and bounded deviations. In both families, the Jordan-curve parameters form a meagre set of full Lebesgue measure.
ER  -