scholarly journals Automorphisms of $\mathbb C^k$ with an invariant non-recurrent attracting Fatou component biholomorphic to $\mathbb C\times (\mathbb C^\ast)^{k-1}$

2020 ◽  
Vol 23 (2) ◽  
pp. 639-666
Author(s):  
Filippo Bracci ◽  
Jasmin Raissy ◽  
Berit Stensønes
Keyword(s):  
Fatou Component

scholarly journals Fatou’s Associates

2020 ◽  
Vol 6 (3-4) ◽  
pp. 459-493
Author(s):  
Vasiliki Evdoridou ◽  
Lasse Rempe ◽  
David J. Sixsmith
Keyword(s):  
Entire Function ◽  
Blaschke Product ◽  
Entire Functions ◽  
Julia Set ◽  
Singular Values ◽  
Strong Relationship ◽  
Inner Function ◽  
Transcendental Entire Function ◽  
Fatou Component ◽  
Finite Blaschke Product

AbstractSuppose that f is a transcendental entire function, $$V \subsetneq {\mathbb {C}}$$ V ⊊ C is a simply connected domain, and U is a connected component of $$f^{-1}(V)$$ f - 1 ( V ) . Using Riemann maps, we associate the map $$f :U \rightarrow V$$ f : U → V to an inner function $$g :{\mathbb {D}}\rightarrow {\mathbb {D}}$$ g : D → D . It is straightforward to see that g is either a finite Blaschke product, or, with an appropriate normalisation, can be taken to be an infinite Blaschke product. We show that when the singular values of f in V lie away from the boundary, there is a strong relationship between singularities of g and accesses to infinity in U. In the case where U is a forward-invariant Fatou component of f, this leads to a very significant generalisation of earlier results on the number of singularities of the map g. If U is a forward-invariant Fatou component of f there are currently very few examples where the relationship between the pair (f, U) and the function g has been calculated. We study this relationship for several well-known families of transcendental entire functions. It is also natural to ask which finite Blaschke products can arise in this manner, and we show the following: for every finite Blaschke product g whose Julia set coincides with the unit circle, there exists a transcendental entire function f with an invariant Fatou component such that g is associated with f in the above sense. Furthermore, there exists a single transcendental entire function f with the property that any finite Blaschke product can be arbitrarily closely approximated by an inner function associated with the restriction of f to a wandering domain.

scholarly journals Fatou maps inℙndynamics

2003 ◽  
Vol 2003 (19) ◽  
pp. 1233-1240 ◽  
Author(s):  
John W. Robertson
Keyword(s):  
Periodic Point ◽  
Complex Projective Space ◽  
Normal Family ◽  
Critical Set ◽  
Fatou Component ◽  
Hyperbolic Fixed Points ◽  
Local Stable Manifold ◽  
Fatou Components ◽  
Complex Projective ◽  
Global Stable Manifolds

We study the dynamics of a holomorphic self-mapfof complex projective space of degreed>1by utilizing the notion of a Fatou map, introduced originally by Ueda (1997) and independently by the author (2000). A Fatou map is intuitively like an analytic subvariety on which the dynamics offare a normal family (such as a local stable manifold of a hyperbolic periodic point). We show that global stable manifolds of hyperbolic fixed points are given by Fatou maps. We further show that they are necessarily Kobayashi hyperbolic and are always ramified byf(and therefore any hyperbolic periodic point attracts a point of the critical set off). We also show that Fatou components are hyperbolically embedded inℙnand that a Fatou component which is attracted to a taut subset of itself is necessarily taut.

Invariant escaping Fatou components with two rank-one limit functions for automorphisms of

2021 ◽  
pp. 1-16
Author(s):  
ANNA MIRIAM BENINI ◽  
ALBERTO SARACCO ◽  
MICHELA ZEDDA
Keyword(s):  
Fatou Component ◽  
Henon Maps ◽  
Hénon Maps ◽  
Generic Rank ◽  
General Growth ◽  
Rank One ◽  
Fatou Components

Abstract We construct automorphisms of ${\mathbb C}^2$ , and more precisely transcendental Hénon maps, with an invariant escaping Fatou component which has exactly two distinct limit functions, both of (generic) rank one. We also prove a general growth lemma for the norm of points in orbits belonging to invariant escaping Fatou components for automorphisms of the form $F(z,w)=(g(z,w),z)$ with $g(z,w):{\mathbb C}^2\rightarrow {\mathbb C}$ holomorphic.

scholarly journals Bounded Fatou Components of Composite Transcendental Entire Functions with Gaps

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Cunji Yang ◽  
Shaoming Wang
Keyword(s):  
Entire Functions ◽  
Fatou Component ◽  
Transcendental Entire Functions ◽  
Fatou Components

We prove that composite transcendental entire functions with certain gaps have no unbounded Fatou component.

scholarly journals A two-dimensional polynomial mapping with a wandering Fatou component

2016 ◽  
Vol 184 (1) ◽  
pp. 263-313 ◽  
Author(s):  
Matthieu Astorg ◽  
Xavier Buff ◽  
Romain Dujardin ◽  
Han Peters ◽  
Jasmin Raissy
Keyword(s):  
Two Dimensional ◽  
Polynomial Mapping ◽  
Fatou Component

scholarly journals Hollow quasi-Fatou components of quasiregular maps

2016 ◽  
Vol 162 (3) ◽  
pp. 561-574
Author(s):  
DANIEL A. NICKS ◽  
DAVID J. SIXSMITH
Keyword(s):  
Entire Function ◽  
Julia Set ◽  
The Other ◽  
Transcendental Entire Function ◽  
Connected Component ◽  
Fatou Component ◽  
Multiply Connected ◽  
Complementary Component ◽  
Fatou Components ◽  
Isolated Point

AbstractWe define a quasi-Fatou component of a quasiregular map as a connected component of the complement of the Julia set. A domain in ℝd is called hollow if it has a bounded complementary component. We show that for each d ⩾ 2 there exists a quasiregular map of transcendental type f: ℝd → ℝd with a quasi-Fatou component which is hollow.Suppose that U is a hollow quasi-Fatou component of a quasiregular map of transcendental type. We show that if U is bounded, then U has many properties in common with a multiply connected Fatou component of a transcendental entire function. On the other hand, we show that if U is not bounded, then it is completely invariant and has no unbounded boundary components. We show that this situation occurs if J(f) has an isolated point, or if J(f) is not equal to the boundary of the fast escaping set. Finally, we deduce that if J(f) has a bounded component, then all components of J(f) are bounded.

scholarly journals Fatou components with punctured limit sets

2014 ◽  
Vol 35 (5) ◽  
pp. 1380-1393 ◽  
Author(s):  
LUKA BOC-THALER ◽  
JOHN ERIK FORNÆSS ◽  
HAN PETERS
Keyword(s):  
Limit Set ◽  
Holomorphic Maps ◽  
Limit Sets ◽  
Fatou Component ◽  
Additional Hypothesis ◽  
Recurrent Case ◽  
Analytic Sets ◽  
Polynomial Endomorphisms ◽  
Fatou Components

We study invariant Fatou components for holomorphic endomorphisms in $\mathbb{P}^{2}$. In the recurrent case these components were classified by Fornæss and Sibony [Classification of recurrent domains for some holomorphic maps. Math. Ann. 301(4) (1995), 813–820]. Ueda [Holomorphic maps on projective spaces and continuations of Fatou maps. Michigan Math J.56(1) (2008), 145–153] completed this classification by proving that it is not possible for the limit set to be a punctured disk. Recently Lyubich and Peters [Classification of invariant Fatou components for dissipative Hénon maps. Preprint] classified non-recurrent invariant Fatou components, under the additional hypothesis that the limit set is unique. Again all possibilities in this classification were known to occur, except for the punctured disk. Here we show that the punctured disk can indeed occur as the limit set of a non-recurrent Fatou component. We provide many additional examples of holomorphic and polynomial endomorphisms of $\mathbb{C}^{2}$ with non-recurrent Fatou components on which the orbits converge to the regular part of arbitrary analytic sets.

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