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For a quasiregular mapping $$f:{\mathbb {R}}^n \rightarrow {\mathbb {R}}^n$$ , with $$n\ge 2$$ , a Julia limiting direction $$\theta \in S^{n-1}$$ arises from a sequence $$(x_n)_{n=1}^{\infty }$$ contained in the Julia set of f, with $$|x_n| \rightarrow \infty $$ and $$x_n/|x_n| \rightarrow \theta $$ . Julia limiting directions have been extensively studied for entire and meromorphic functions in the plane. In this paper, we focus on Julia limiting directions in higher dimensions. First, we give conditions under which every direction is a Julia limiting direction. Our methods show that if a quasi-Fatou component contains a sectorial domain, then there is a polynomial bound on the growth in the sector. Second, we give a sufficient, but not necessary, condition in $${\mathbb {R}}^3$$ for a set $$E\subset S^2$$ to be the set of Julia limiting directions for a quasiregular mapping. The methods here will require showing that certain sectorial domains in $${\mathbb {R}}^3$$ are ambient quasiballs. This is a contribution to the notoriously hard problem of determining which domains are the image of the unit ball $${\mathbb {B}}^3$$ under an ambient quasiconformal mapping of $${\mathbb {R}}^3$$ onto itself.

The goal of this article is to study a rigidity property of Julia sets of certain classes of automorphisms in $$\mathbb C^k$$ , $$k \ge 3.$$ First, we study the relation between two polynomial shift-like maps in $$\mathbb C^k$$ , $$k \ge 3$$ , that share the same forward and backward Julia sets. Secondly, we study the relation between any pair of skew products of Henon maps in $$\mathbb C^3$$ having the same (forward and backward) Julia sets. Also in the same spirit, we further establish a similar relation between skew products of Henon maps fibered over a compact metric space, sharing the same Julia sets.

In 1988, Mayer proved the remarkable fact that infinity is an explosion point for the set of endpoints of the Julia set of an exponential map that has an attracting fixed point. That is, the set is totally separated (in particular, it does not have any nontrivial connected subsets), but its union with the point at infinity is connected. Answering a question of Schleicher, we extend this result to the set of "escaping endpoints" in the sense of Schleicher and Zimmer, for any exponential map for which the singular value belongs to an attracting or parabolic basin, has a finite orbit, or escapes to infinity under iteration (as well as many other classes of parameters). Furthermore, we extend one direction of the theorem to much greater generality, by proving that the set of escaping endpoints joined with infinity is connected for any transcendental entire function of finite order with bounded singular set. We also discuss corresponding results for *all* endpoints in the case of exponential maps; in order to do so, we establish a version of Thurston's "no wandering triangles" theorem., Comment: 35 pages. To appear in Comput. Methods Funct. Theory. V2: Authors' final accepted manuscript. Revisions and clarifications have been made throughout from V1. This includes improvements in the proof of Proposition 6.11 and Theorem 8.1, as well as corrections in Remarks 7.1 and 7.3 concerning differing definitions of escaping endpoints in greater generality

In this paper we discuss the dynamics as well as the structure of the parameter space of the one-parameter family of rational maps $${f_{t}(z)}= - {t\over 4} {(z^{2}- 2)^{2}\over {z^{2}- 1}}$$ with free critical orbit $$\pm\sqrt{2}\mathop \rightarrow \limits^{(2)}0 \mathop \rightarrow \limits^{(4)}t \mathop\rightarrow \limits^{(1)}\cdots.$$ In particular we show that for any escape parameter t, the boundary of the basin at infinity A t is either a Cantor set, a curve with infinitely many complementary components, or else a Jordan curve. In the latter case the Julia set is a Sierpinski curve.

We survey some results on the iteration of quasiregular mappings. In particular we discuss some recent results on the dynamics of quasiregular maps which are not uniformly quasiregular.

It is known that if f is a meromorphic map of the complement of three points on the extended complex plane into itself, then f is a Mobius map. We consider which subdomains of the extended complex plane also have this property.

Periodic components of the Fatou set of a rational map are either simply, doubly or infinitely connected. However, preimages of these may possibly possess a higher finite connectivity. The purpose of this article is to study certain classes of generalized Blaschke products that possess Fatou components of a given arbitrary finite connectivity number.

In this note we prove that the so-called Sierpi\’nski holes in the parameter plane 0 < ¦λ¦ < ∞ of the McMullen family Fλ(z) = z m + λ/z l, m ≥ 2 and l ≥ 1 fixed, are simply connected, and determine the total number of these domains.

The main purpose of the Fatou-Julia theory is to study the global behaviour of the sequence (fn) of iterates of a rational function f. In this survey article we consider generalized iteration which means that the iterated function f may vary from step to step. More precisely, let (fn) be a sequence of rational functions, and let \( F_{n}: = f_{n}\circ \cdots \circ f_{1}\) be the sequence of forward compositions, and let the Fatou set and Julia set of (Fn) be defined as usual. Then, in general, most of the results of the Fatou-Julia theory fail to hold. On the other hand, under appropriate restrictions on the sequence (fn) many results can be carried over to this more general situation, but the proofs are often completely different.

For a polynomial P it is well known that its Julia set % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A! ${\cal J_P}$ is connected if and only if the orbits of the finite critical points are bounded. But there is no such simple criteria for the connectedness of the Julia set of a rational function. Indeed, up to the very nice result of Shishikura that any rational function which has one repelling fixed point only has a connected Julia set almost nothing is known on the connectivity. In the first part of the paper we give constructive sufficient conditions for a basin of attraction to be completely invariant and the Julia set to be connected. Then it is shown that the connectedness of a basin of attraction depends heavily on the fact whether the critical points from the basin tend to the attracting fixed point z 0 via a preimage of z 0 or not. As a consequence we obtain for instance that rational functions with a finite postcritical set or with a Fatou set which contains no Herman rings and each component of which contains at most one critical point, counted without multiplicity, have a connected Julia set.

In this paper we consider rational functions \(f: \overline {\rm C} \rightarrow \overline {\rm C}\) with parabolic and critical points contained in their Julia sets J(f) such that $$\sum^{\infty}_{n = 1}|(f^{n})^\prime(f(c))|^{-1}< \infty$$ for each critical point c ∈ J(f). We calculate the Hausdorff dimensions of subsets of J(f) consisting of elements z for which $$\inf \left\{ {dist\left( {f^n \left( z \right), Crit\left( f \right)} \right):n \geqslant 0} \right\} > 0$$ and which are well-approximable by backward iterates of the parabolic periodic points of f.