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.
In this paper, we prove that the ratio of the modulus of the iterates of two points in an escaping Fatou component may be bounded even if the orbit of the component contains an infinite modulus annulus sequence and this case cannot happen when the maximal modulus of the meromorphic function is large enough. Therefore, we extend the related results for entire functions to ones for meromorphic functions with infinitely many poles. And we investigate the fast escaping Fatou components of meromorphic functions defined in [44] in terms of the Nevanlinna characteristic instead of the maximal modulus…
We show that for any transcendental meromorphic function $f$ there is a point $z$ in the Julia set of $f$ such that the iterates $f^n(z)$ escape, that is, tend to $\infty$, arbitrarily slowly. The proof uses new covering results for analytic functions. We also introduce several slow escaping sets, in each of which $f^n(z)$ tends to $\infty$ at a bounded rate, and establish the connections between these sets and the Julia set of $f$. To do this, we show that the iterates of $f$ satisfy a strong distortion estimate in all types of escaping Fatou components except one, which we call a plane-filli…
This article concerns the iteration of quasiregular mappings on $\mathbb{R}^d$ and entire functions on $\mathbb{C}$. It is shown that there are always points at which the iterates of a quasiregular map tend to infinity at a controlled rate. Moreover, an asymptotic rate of escape result is proved that is new even for transcendental entire functions. Let $f:\mathbb{R}^d\to\mathbb{R}^d$ be quasiregular of transcendental type. Using novel methods of proof, we generalise results of Rippon and Stallard in complex dynamics to show that the Julia set of $f$ contains points at which the iterates $f^n$ …
We show that for any quasimeromorphic mapping with an essential singularity at infinity, there exist points whose iterates tend to infinity arbitrarily slowly. This extends a result by Nicks for quasiregular mappings, and Rippon and Stallard for transcendental meromorphic functions on the complex plane. We further establish a new result for the growth rate of quasiregular mappings near an essential singularity, and briefly extend some results regarding the bounded orbit set and the bungee set to the quasimeromorphic setting.
Much recent work on the iterates of a transcendental entire function $f$ has been motivated by Eremenko's conjecture that all the components of the escaping set $I(f)$ are unbounded. Here we show that if $I(f)$ is disconnected, then the set $I(f)\setminus D$ has uncountably many unbounded components for any open disc $D$ that meets the Julia set of $f$. For the set $A_R(f)$, which is the `core' of the fast escaping set, we prove the much stronger result that for some $R>0$ either $A_R(f)$ is connected and has the structure of an infinite spider's web or it has uncountably many components each …
We study topological properties of the escaping endpoints and fast escaping endpoints of the Julia set of complex exponential $\exp(z)+a$ when $a\in (-\infty,-1)$. We show neither space is homeomorphic to the whole set of endpoints. This follows from a general result stating that for every transcendental entire function $f$, the escaping Julia set $I(f)\cap J(f)$ is first category.
We study the dynamics of iterated cosine maps $E\colon z \mapsto ae^z+be^{-z},$ with $a,b \in \C\setminus \{0\}$. We show that the points which converge to infinity under iteration are organized in the form of rays and, as in the exponential family, every escaping point is either on one of these rays or the landing point of a unique ray. Thus we get a complete classification of the escaping points of the cosine family, confirming a conjecture of Eremenko in this case. We also get a particularly strong version of the ``dimension paradox'': the set of rays has Hausdorff dimension 1, while the se…
Let $f$ and $g$ be permutable transcendental entire functions. We use a recent analysis of the dynamical behaviour in multiply connected wandering domains to make progress on the long standing conjecture that the Julia sets of $f$ and $g$ are equal; in particular, we show that $J(f)=J(g)$ provided that neither $f$ nor $g$ has a simply connected wandering domain in the fast escaping set.
We consider the transcendental entire function $ f(z)=z+e^{-z} $, which has a doubly parabolic Baker domain $U$ of degree two, i.e. an invariant stable component for which all iterates converge locally uniformly to infity, and for which the hyperbolic distance between successive iterates converges to zero. It is known from general results that the dynamics on the boundary is ergodic and recurrent and that the set of points in $\partial U$ whose orbit escapes to infity has zero harmonic measure. For this model we show that stronger results hold, namely that this escaping set is non-empty, it is…
Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学物理学院）；Faculty of Mathematics and Computer Science, University of Barcelona（巴塞罗那大学数学与计算机科学学院）；Institute for Mathematics, Physics and Mechanics（数学、物理与力学研究所）
We show that for many families of transcendental entire functions $f$ the property that $m^n(r)\to\infty$ as $n\to \infty$, for some $r>0$, where $m(r)=\min\{|f(z)|:|z|=r\}$, implies that the escaping set $I(f)$ of $f$ has the structure of a spider's web. In particular, in this situation $I(f)$ is connected, so Eremenko's conjecture holds. We also give new examples of families of functions for which this iterated minimum modulus condition holds and new families for which it does not hold.
For a polynomial p with a repelling fixed point w, we consider Poincaré functions of p at w, i.e. entire functions L which satisfy L(0)=w and p(L(z))=L(p'(w)*z) for all z in the complex plane. We show that if the component of the Julia set of p that contains w equals {w}, then the (fast) escaping set of L is a spider's web; in particular it is connected. More precisely, we classify all linearizers of polynomials with regards to the spider's web structure of the set of all points which escape faster than the iterates of the maximum modulus function at a sufficiently large point.
In this paper we study two classes of meromorphic functions previously studied by Mayer, Kotus, and Urbański. In particular we estimate a lower bound for the Julia set and the set of escaping points for non-autonomous additive and affine perturbations of functions from these classes. For particular classes we are able to calculate these dimensions exactly. In these cases, we are able to reinterpret our results to show that the Hausdorff dimension of the set of points in the Julia set which escape to infinity is stable under sufficiently small additive perturbations. We accomplish this by const…
Let a>0, F: R^2 -> R^2 be a differentiable (not necessarily C^1) map and Spec(F) be the set of (complex) eigenvalues of the derivative F'(p) when p varies in R^2. (a) If Spec(F) is disjoint of the interval [1,1+a[, then Fix(F) has at most one element, where Fix(F) denotes the set of fixed points of F. (b) If Spec(F) is disjoint of the real line R, then Fix(F^2) has at most one element. (c) If F is a C^1 map and, for all p belonging to R^2, the derivative F'(p) is neither a homothety nor has simple real eigenvalues, then Fix(F^2) has at most one element, provided that Spec(F) is disjoint of eit…
Let $q$ be a prime power and $ϕ$ a rational function with coefficients in a finite field $\mathbb{F}_q$. For $n \geq 1$, each element of $\mathbb{P}^1(\F_{q^n})$ is either periodic or strictly preperiodic under iteration of $ϕ$. Denote by $a_n$ the proportion of periodic elements. Little is known about how $a_n$ changes as $n$ grows, unless $ϕ$ is a power map or Chebyshev polynomial. We give the first results on this question for a wider class of rational functions: $a_n$ has lim inf $0$ when $q$ is odd and $ϕ$ is quadratic and neither Lattès nor conjugate to a one-parameter family of exceptio…
