Punctured parabolic cylinders in automorphisms of $\mathbb{C}^{2}$

DOI10.1093/IMRN/RNAA217arXiv1909.00765MaRDI QIDQ6324552

Publication date: 2 September 2019

Abstract: We show the existence of automorphisms

F

of

m a t h b b C^{2}

with a non-recurrent Fatou component

O m e g a

biholomorphic to

m a t h b b C i m e s m a t h b b C^{*}

that is the basin of attraction to an invariant entire curve on which

F

acts as an irrational rotation. We further show that the biholomorphism

O m e g a o m a t h b b C i m e s m a t h b b C^{*}

can be chosen such that it conjugates

F

to a translation

(z, w) m a p s t o (z + 1, w)

, making

O m e g a

a parabolic cylinder as recently defined by L.~Boc Thaler, F.~Bracci and H.~Peters.

F

and

O m e g a

are obtained by blowing up a fixed point of an automorphism of

m a t h b b C^{2}

with a Fatou component of the same biholomorphic type attracted to that fixed point, established by F.~Bracci, J.~Raissy and B.~Stens{o}nes. A crucial step is the application of the density property of a suitable Lie algebra to show that the automorphism in their work can be chosen such that it fixes a coordinate axis. We can then remove the proper transform of that axis from the blow-up to obtain an

F

-stable subset of the blow-up that is biholomorphic to

m a t h b b C^{2}

. Thus we can interpret

F

as an automorphism of

m a t h b b C^{2}

.

Mathematics Subject Classification ID

Small divisors, rotation domains and linearization in holomorphic dynamics (37F50) Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables (32H02) Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets (37F10) Normal families of holomorphic functions, mappings of several complex variables, and related topics (taut manifolds etc.) (32A19) Combinatorics and topology in relation with holomorphic dynamical systems (37F20) Higher-dimensional holomorphic and meromorphic dynamics (37F80)

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