The wandering domain problem for attracting polynomial skew products

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.

A Baker omitted value, in short bov of a transcendental meromorphic function f is an omitted value such that there is a disk D centered at the bov for which each component of the boundary of \(f^{-1}(D)\) is bounded. Assuming all the iterates \(f^n\) to be analytic in a neighborhood of its bov, this article proves a number of results on the rotation domains of the function. The number of p-periodic Herman rings is shown to be finite for each \(p \ge 1\). It is also proved that every Julia component intersects the boundaries of at most finitely many Herman rings. Further, if the bov is the only limit point of the critical values then it is shown that f has infinitely many repelling fixed points. If a repelling periodic point of period p is on the boundary of a p-periodic rotation domain then the periodic point is shown to be on the boundary of infinitely many Fatou components. As a corollary we have shown that if D is a p-periodic rotation domain of f and \(f^p\) is univalent in a neighbourhood of the boundary \( \partial D \) of D then f has no repelling p-periodic point on \( \partial D \). Under additional assumptions on the critical points, a sufficient condition is found for a Julia component to be a singleton. As a consequence, it is proved that if the boundary of a wandering domain W accumulates at some point of the plane under the iteration of f then each limit of \(f^n\) on W is either a parabolic periodic point or in the \(\omega \)-limit set of recurrent critical points. Using the same ideas, the boundaries of rotation domains are shown to be in the \(\omega \)-limit set of recurrent critical points.

Let M be the class of all transcendental meromorphic functions f : ℂ → ℂ ∪ {∞} with at least two poles or one pole that is not an omitted value, and Mo = {f ∈ M : f has at least one omitted value}. Some dynamical issues of the functions in Mo are addressed in this article. A complete classification in terms of forward orbits of all the multiply connected Fatou components is made. As a corollary, it follows that the Julia set is not totally disconnected unless all the omitted values are contained in a single Fatou component. Non-existence of both Baker wandering domains and invariant Herman rings are proved. Eventual connectivity of each wandering domain is proved to exist. For functions with exactly one pole, we show that Herman rings of period 2 also do not exist. A necessary and sufficient condition for the existence of a dense subset of singleton buried components in the Julia set is established for functions with two omitted values. The conjecture that a meromorphic function has at most two completely invariant Fatou components is confirmed for all f ∈ Mo except in the case when f has a single omitted value, no critical value and is of infinite order. Some relevant examples are discussed.

We present several results on the compactness of the space of morphisms between analytic spaces in the sense of Berkovich. We show that under certain conditions on the source, every sequence of analytic maps having an affinoid target has a subsequence that converges pointwise to a continuous map. We also study the class of continuous maps that arise in this way. Locally, they turn to be analytic after a certain base change. Our results naturally lead to a definition of normal families. We give some applications to the dynamics of an endomorphism of the projective space. We introduce two natural notions of Fatou set and generalize to the non-Archimedan setting a theorem of Ueda stating that every Fatou component is hyperbolically imbedded in the projective space.

We study the dynamics of a holomorphic self‐map f of complex projective space of degree d > 1 by 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 of f are 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 by f (and therefore any hyperbolic periodic point attracts a point of the critical set of f). We also show that Fatou components are hyperbolically embedded in ℙn and that a Fatou component which is attracted to a taut subset of itself is necessarily taut.

Dynamics of composition of entire functions is well related to it's factors, as it is known that for entire functions f and g, fog has wandering domain if and only if gof has wandering domain. However the Fatou components may have different structures and properties. In this paper we have shown the existence of domains with all possibilities of wandering and periodic in given angular region θ.

We consider polynomial maps of the form f ( z , w ) = ( p ( z ) , q ( z , w ) ) f(z,w) = (p(z),q(z,w)) that extend as holomorphic maps of C P 2 \mathbb {CP}^2 . Mattias Jonsson introduces in “Dynamics of polynomial skew products on C 2 \mathbf {C}^2 ” [Math. Ann., 314(3): 403–447, 1999] a notion of connectedness for such polynomial skew products that is analogous to connectivity for the Julia set of a polynomial map in one-variable. We prove the following dichotomy: if f f is an Axiom-A polynomial skew product, and f f is connected, then every Fatou component of f f is homeomorphic to an open ball; otherwise, some Fatou component of F F has infinitely generated first homology.

In this paper, we investigate the dynamics of a broader class of functions which are meromorphic outside a compact totally disconnected set. We shall establish the connections between the Fatou components and the singularities of the inverse function and, accordingly, give sufficient conditions for the non-existence of wandering domains or Baker domains, and for the Julia set to be the Riemann sphere. Through the discussion of permutability of such functions, we shall construct several transcendental meromorphic functions which have Baker domains and wandering domains with special properties; for example, wandering and Baker domains with a critical value on the boundary and a wandering domain with the boundary being a Jordan curve (some such examples for entire functions were exhibited in other papers) and those of non-finite type which have no wandering domains.

Baker's conjecture states that a transcendental entire function of order less than $1/2$ has no unbounded Fatou components. It is known that, for such functions, there are no unbounded periodic Fatou components and so it remains to show that they can also have no unbounded wandering domains. Here we introduce completely new techniques to show that the conjecture holds in the case that the transcendental entire function is real with only real zeros, and we prove the much stronger result that such a function has no orbits of unbounded wandering domains whenever the order is less than 1. This raises the question as to whether such wandering domains can exist for any transcendental entire function with order less than 1. Key ingredients of our proofs are new results in classical complex analysis with wider applications. These new results concern: the winding properties of the images of certain curves proved using extremal length arguments, growth estimates for entire functions, and the distribution of the zeros of entire functions of order less than 1.

Let f(z) be a transcendental meromorphic function. The paper investigates, using the hyperbolic metric, the relation between the forward orbit P(f) of singularities of f−1 and limit functions of iterations of f in its Fatou components. It is mainly proved, among other things, that for a wandering domain U, all the limit functions of {fn|U} lie in the derived set of P(f) and that if fnp|V→ q(n→ +∞) for a Fatou component V, then either q is in the derived set of Sp (f) or fp(q) = q. As applications of main theorems, some sufficient conditions of the non-existence of wandering domains and Baker domains are given.

We shall show that for certain holomorphic maps, all Fatou components are simply connected. We also discuss the relation between wandering domains and singularities for certain meromorphic maps.

Let f f be a transcendental entire function and U U be a Fatou component of f f . We show that if U U is an escaping wandering domain of f f , then most boundary points of U U (in the sense of harmonic measure) are also escaping. In the other direction we show that if enough boundary points of U U are escaping, then U U is an escaping Fatou component. Some applications of these results are given; for example, if I ( f ) I(f) is the escaping set of f f , then I ( f ) ∪ { ∞ } I(f)\cup \{\infty \} is connected.

For a transcendental entire function f, we study the set of points BU(f) whose iterates under f neither escape to infinity nor are bounded. We give new results on the connectedness properties of this set and show that, if U is a Fatou component that meets BU(f), then most boundary points of U (in the sense of harmonic measure) lie in BU(f). We prove this using a new result concerning the set of limit points of the iterates of f on the boundary of a wandering domain. Finally, we give some examples to illustrate our results.

Suppose that f is a transcendental entire function, V subsetneq {mathbb {C}} is a simply connected domain, and U is a connected component of f^{-1}(V). Using Riemann maps, we associate the map f :U rightarrow V to an inner function g :{mathbb {D}}rightarrow {mathbb {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.

Observations of high-redshift quasars imply the presence of supermassive black holes (SMBHs) already at $z$ ∼ 7.5. An appealing and promising pathway to their formation is the direct collapse scenario of a primordial gas in atomic-cooling haloes at $z$ ∼ 10–20, when the $\rm H_{2}$ formation is inhibited by a strong background radiation field, whose intensity exceeds a critical value, Jcrit. To estimate Jcrit, typically, studies have assumed idealized spectra, with a fixed ratio of $\rm H_{2}$ photodissociation rate $k_{\rm H_2}$ to the $\rm H^-$ photodetachment rate $k_{\rm H^-}$. This assumption, however, could be too narrow in scope as the nature of the background radiation field is not known precisely. In this work we argue that the critical condition for suppressing the H2 cooling in the collapsing gas could be described in a more general way by a combination of $k_{\rm H_2}$ and $k_{\rm H^-}$ parameters, without any additional assumptions about the shape of the underlying radiation spectrum. By performing a series of cosmological zoom-in simulations covering a wide range of relevant $k_{\rm H_2}$ and $k_{\rm H^-}$ parameters, we examine the gas flow by following evolution of basic parameters of the accretion flow. We test under what conditions the gas evolution is dominated by $\rm H_{2}$ and/or atomic cooling. We confirm the existence of a critical curve in the $k_{\rm H_2}{\!-\!}k_{\rm H^-}$ plane and provide an analytical fit to it. This curve depends on the conditions in the direct collapse, and reveals domains where the atomic cooling dominates over the molecular cooling. Furthermore, we have considered the effect of $\rm H_{2}$ self-shielding on the critical curve, by adopting three methods for the effective column density approximation in $\rm H_{2}$. We find that the estimate of the characteristic length scale for shielding can be improved by using λJeans25, which is 0.25 times that of the local Jeans length, which is consistent with previous one-zone modelling.