Related papers: Ruelle operator and transcendental entire maps

In this note, we give an elementary proof of the absence of invariant line fields on the conical Julia set of an analytic function of one variable. This proof applies not only to rational as well as transcendental meromorphic functions…

Let $C_v$ be a complete, algebraically closed non-archimedean field, and let $f \in C_v(z)$ be a rational function of degree $d \geq 2$. If $f$ satisfies a bounded contraction condition on its Julia set, we prove that small perturbations of…

Let $f$ be a rational map with degree at least two. We prove that $f$ has at least $2$ disjoint and infinite critical orbits in the Julia set if it has a Herman ring. This result is sharp in the following sense: there exists a cubic…

Let $f$ be an entire transcendental function of finite order and $\Delta$ be a forward invariant bounded Siegel disk for $f$ with rotation number in Herman's class $\mathcal{H}$. We show that if $f$ has two singular values with bounded…

Let $f$ and $g$ be two quasiregular maps in $\mathbb{R}^d$ that are of transcendental type and also satisfy $f\circ g =g \circ f$. We show that if the fast escaping sets of those functions are contained in their respective Julia sets then…

Let $f$ be a polynomial-like mapping of the sphere of degree $d \geq 2$. We show that the Julia set $J(f)$ of $f$ cannot be the union of a finite number of proper indecomposable subcontinua. As a corollary, we prove that $J(f)$ is an…

A transcendental entire function f is called geometrically finite if the intersection of the set of singular values with the Fatou set is compact and the intersection of the postsingular set with the Julia set is finite. (In particular,…

For polynomials, local connectivity of Julia sets is a much-studied and important property. Indeed, when the Julia set of a polynomial of degree $d\geq 2$ is locally connected, the topological dynamics can be completely described as a…

We analyze the infinitesimal effect of holomorphic perturbations of the dynamics of a structurally stable rational map on a neighborhood of its Julia set. This implies some restrictions on the behavior of critical points.

We use Bowen's definition of topological entropy and Ahlfors five islands theorem, as well as the theory of polynomial-like mappings, to show that the topological entropy of any entire transcendental function is infinity. In addition the…

In this paper, we show the existence of a transcendental function $f\in\mathbb{Z}\{z\}$ with coefficients that are almost all bounded such that $f$ and all its derivatives assume algebraic values at algebraic points. Furthermore, we…

In this paper, we study the large scaled geometric structure of Julia sets of entire and meromorphic functions. Roughly speaking, the structure gives us some asymptotic information about the Julia set near the essential singularity. We will…

We show that the iterative logarithm of each non-linear entire function is differentially transcendental over the ring of entire functions, and we give a sufficient criterion for such an iterative logarithm to be differentially…

By investigating which level of universality composition operators $C_f$ can have, where the symbol $f$ is given by the restriction of a transcendental entire function to suitable parts of the Fatou set of $f$, this work combines the theory…

In this paper, we prove that for any post-critically finite rational map $f$ on the Riemann sphere $\overline{\mathbb{C}}$, and for each sufficiently large integer $n$, there exists a finite and connected graph $G$ in the Julia set of $f$…

We prove the existence of transcendental entire functions $f$ having a property studied by Mahler, namely that $f(\overline{\mathbb{Q}})\subseteq \overline{\mathbb{Q}}$ and $f^{-1}(\overline{\mathbb{Q}})\subseteq \overline{\mathbb{Q}}$, and…

We study Nevanlinna functions f that are transcendental meromorphic functions having N asymptotic values and no critical values. In [KK] it was proved that if the orbits of all the asymptotic values have accumulation sets that are compact…

The no invariant line fields conjecture is one of the main outstanding problems in traditional complex dynamics. In this paper we consider non-autonomous iteration where one works with compositions of sequences of polynomials with suitable…

We exhibit an analytic family of hyperbolic, even disjoint type, entire functions for which the hyperbolic dimension does not vary analytically. Additionally we answer several questions in thermodynamic formalism of entire functions such as…

We survey recent work of Burghelea, Malta and both authors on the topology of critical sets of nonlinear ordinary differential operators. For a generic nonlinearity $f$, the critical set of the first order nonlinear operator $F_1(u)(t) =…