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On the rank-one limit sets for maps in the constant-Jacobian shear group of ℂ2

  • Luka Boc Thaler
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Abstract

Let 𝒮c be the subgroup of Aut(ℂ2) generated by affine maps and shear automorphisms with constant Jacobian. We determine the possible conformal types of the rank-one limit set of an invariant recurrent Fatou component when the induced dynamics has infinite order. In particular we show that, under this infinite-order hypothesis, the limit set of a finite composition of Hénon maps (polynomial or transcendental), can only be conformally equivalent to a disk or an annulus. Thus we exclude the remaining three conformal types, i.e. 𝔻*, ℂ* and ℂ, from the classification result by Arosio, Benini, Fornæss and Peters. We also prove that the set of fixed points of a finite composition of Hénon maps (polynomial or transcendental) contains no entire curves.

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Cite

BibTeXRIS
BibTeX
@misc{thaler2026rankone,
  title = {{On the rank-one limit sets for maps in the constant-Jacobian shear group of $\mathbbC^2$}},
  author = {Luka Boc Thaler},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.17335},
  eprint = {2610.17335},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.17335},
}
RIS
TY  - UNPB
TI  - On the rank-one limit sets for maps in the constant-Jacobian shear group of ℂ^2
AU  - Luka Boc Thaler
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.17335
UR  - https://arxiv.org/abs/2610.17335
AB  - Let 𝒮_c be the subgroup of Aut(ℂ^2) generated by affine maps and shear automorphisms with constant Jacobian. We determine the possible conformal types of the rank-one limit set of an invariant recurrent Fatou component when the induced dynamics has infinite order. In particular we show that, under this infinite-order hypothesis, the limit set of a finite composition of Hénon maps (polynomial or transcendental), can only be conformally equivalent to a disk or an annulus. Thus we exclude the remaining three conformal types, i.e. 𝔻^*, ℂ^* and ℂ, from the classification result by Arosio, Benini, Fornæss and Peters. We also prove that the set of fixed points of a finite composition of Hénon maps (polynomial or transcendental) contains no entire curves.
ER  -