A transcendental Hénon map with an oscillating wandering Short $${\mathbb {C}}^2$$

Arosio, Leandro; Thaler, Luka Boc; Peters, Hans

doi:10.1007/s00209-020-02677-4

Cited by 4 publications

(2 citation statements)

References 10 publications

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“…Much like Fatou-Bieberbach domains, Short C 2 's come in a variety of shapes and sizes, and possess a number of intriguing properties as can be seen from the examples in [1], [5], [6], [7], [8] and [13]. The purpose of this paper is to study the holomorphic automorphism group of Short C 2 's that arise in (ii) above.…”

Section: Introductionmentioning

confidence: 99%

“…Bounded forward orbits are therefore eventually trapped in V R ∪ V − R , and the unbounded ones approach infinity while remaining in V + R . Let 1) shows that if H n (x, y) = (x n , y n ), the degree of x n is strictly dominated by the degree of y n for all n ≥ 1 and that y n+1 ∼ y d n . Hence, the limit (which defines the Green's function)…”

Section: Introductionmentioning

confidence: 99%

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On the automorphism group of certain Short $\mathbb C^2$'s

Bera¹,

Pal²,

Verma³

2021

Preprint

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For a Hénon map of the form H(x, y) = (y, p(y) − ax), where p is a polynomial of degree at least two and a = 0, it is known that the sub-level sets of the Green's function G + H associated with H are Short C 2 's. For a given c > 0, we study the holomorphic automorphism group of such a Short C 2 , namely Ωc = {G + H < c}. The unbounded domain Ωc ⊂ C 2 is known to have smooth real analytic Levi-flat boundary. Despite the fact that Ωc admits an exhaustion by biholomorphic images of the unit ball, it turns out that its automorphism group, Aut(Ωc) cannot be too large. On the other hand, examples are provided to show that these automorphism groups are non-trivial in general. We also obtain necessary and sufficient conditions for such a pair of Short C 2 's to be biholomorphic.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the automorphism group of certain Short $\mathbb C^2$'s

Bera¹,

Pal²,

Verma³

2021

Preprint

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Notes on the Short $${\mathbb {C}}^k$$’s

Fornæss¹,

Pal

2022

J Geom Anal

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Emergence of wandering stable components

Berger

Biebler

2022

J. Amer. Math. Soc.

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We prove the existence of a locally dense set of real polynomial automorphisms of C 2 \mathbb C^2 displaying a wandering Fatou component; in particular this solves the problem of their existence, reported by Bedford and Smillie in 1991. These Fatou components have non-empty real trace and their statistical behavior is historic with high emergence. The proof is based on a geometric model for parameter families of surface real mappings. At a dense set of parameters, we show that the dynamics of the model displays a historic, high emergent, stable domain. We show that this model can be embedded into families of Hénon maps of explicit degree and also in an open and dense set of 5 5 -parameter C r C^r -families of surface diffeomorphisms in the Newhouse domain, for every 2 ≤ r ≤ ∞ 2\le r\le \infty and r = ω r=\omega . This implies a complement of the work of Kiriki and Soma [Adv. Math. 306 (2017), pp. 524–588], a proof of the last Taken’s problem in the C ∞ C^{\infty } and C ω C^\omega -case. The main difficulty is that here perturbations are done only along finite-dimensional parameter families. The proof is based on the multi-renormalization introduced by Berger [Zoology in the Hénon family: twin babies and Milnor’s swallows, 2018].

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A transcendental Hénon map with an oscillating wandering Short $${\mathbb {C}}^2$$

Cited by 4 publications

References 10 publications

On the automorphism group of certain Short $\mathbb C^2$'s

On the automorphism group of certain Short $\mathbb C^2$'s

Notes on the Short $${\mathbb {C}}^k$$’s

Emergence of wandering stable components

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