Bounded vertical deviations and irrational circle factors in Dehn Twist classes

Abstract

We prove that a homeomorphism of the two-torus in a Dehn twist homotopy class is a topological extension of an irrational circle rotation if and only if its vertical rotation set reduces to a single irrational number and the homeomorphism exhibits uniformly bounded rotational deviations in the vertical direction. This result has been previously obtained by the second author under an additional hypothesis about the geometry of the non-wandering set. We show that this hypothesis can be entirely dropped: the large-scale shear intrinsic to a Dehn twist eliminates the obstructions coming from the wandering set, which cannot be removed in the homotopy class of the identity. Finally, we prove that the factor is unique up to a circle rotation and is locally constant on the wandering set. Consequently, every connected component of the wandering set is contained in a single fiber, is itself a wandering domain and is either inessential or annular.

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BibTeX RIS

BibTeX

@misc{correa2026bounded,
  title = {{Bounded vertical deviations and irrational circle factors in Dehn Twist classes}},
  author = {Heric Corrêa and Alejandro Kocsard},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.11627},
  eprint = {2609.11627},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11627},
}

RIS

TY  - UNPB
TI  - Bounded vertical deviations and irrational circle factors in Dehn Twist classes
AU  - Heric Corrêa
AU  - Alejandro Kocsard
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.11627
UR  - https://arxiv.org/abs/2609.11627
AB  - We prove that a homeomorphism of the two-torus in a Dehn twist homotopy class is a topological extension of an irrational circle rotation if and only if its vertical rotation set reduces to a single irrational number and the homeomorphism exhibits uniformly bounded rotational deviations in the vertical direction. This result has been previously obtained by the second author under an additional hypothesis about the geometry of the non-wandering set. We show that this hypothesis can be entirely dropped: the large-scale shear intrinsic to a Dehn twist eliminates the obstructions coming from the wandering set, which cannot be removed in the homotopy class of the identity. Finally, we prove that the factor is unique up to a circle rotation and is locally constant on the wandering set. Consequently, every connected component of the wandering set is contained in a single fiber, is itself a wandering domain and is either inessential or annular.
ER  -

Abstract

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