Related papers: Fatou's web
In 1909, Hardy gave an example of a transcendental entire function, $f$, with the property that the set of points where $f$ achieves its maximum modulus, $\mathcal{M}(f)$, has infinitely many discontinuities. This is one of only two known…
For a set $x$, let $\mathcal{S}(x)$ be the set of all permutations of $x$. We study several aspects of this notion in $\mathsf{ZF}$. The main results are as follows: (1) $\mathsf{ZF}$ proves that for all sets $x$, if $\mathcal{S}(x)$ is…
Given any dimension function $h$, we construct a perfect set $E \subseteq \mathbb{R}$ of zero $h$-Hausdorff measure, that contains any finite polynomial pattern. This is achieved as a special case of a more general construction in which we…
We prove that there exists a non-trivial transcendental semigroup S such that the wandering (or pre-periodic or periodic) components of Fatou set F(S) has at least a simply connected domain D.
We introduce a new class of entire functions $\mathcal{E}$ which consists of all $F_0\in\mathcal{O}(\mathbb{C})$ for which there exists a sequence $(F_n)\in \mathcal{O}(\mathbb{C})$ and a sequence $(\lambda_n)\in\mathbb{C}$ satisfying…
Let f be a transcendental meromorphic function. Suppose that the finite part of the postsingular set of f is bounded, that f has no recurrent critical points or wandering domains, and that the degree of pre-poles of f is uniformly bounded.…
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 show that an invariant Fatou component of a hyperbolic transcendental entire function is a bounded Jordan domain (in fact, a quasidisc) if and only if it contains only finitely many critical points and no asymptotic curves. We use this…
Guided by classical concepts, we define the notion of \emph{ends} of an iterated function system and prove that the number of ends is an upper bound for the number of nondegenerate components of its attractor. The remaining isolated points…
We show that a rational function $f$ of degree $>1$ on the projective line over an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value has no potentially good reductions if and only…
We develop techniques that lay out a basis for generalizations of the famous Thurston's Topological Characterization of Rational Functions for an infinite set of marked points and branched coverings of infinite degree. Analogously to the…
For the family of complex rational functions known as "Generalized McMullen maps", F(z) = z^n + a/z^n+b, for complex parameters a and b, with a nonzero, and any integer n at least 3 fixed, we reveal, and provide a combinatorial model for,…
We show that if the maximum modulus of a quasiregular mapping f grows sufficiently rapidly then there exists a non-empty escaping set I(f) consisting of points whose forward orbits under iteration tend to infinity. This set I(f) has an…
We prove that all entire transcendental entire functions have infinite topological entropy.
If $f$ is a transcendental entire function with only algebraic singularities we calculate the Ruelle operator of $f$. Moreover, we prove both (i) if $f$ has a summable critical point, then $f$ is not structurally stable under certain…
We prove that entire transcendental holomorphic functions with an omitted value have infinite entropy. A proof for general transcendental entire functions will be given in an upcoming paper.
We give conditions ensuring that the Julia set and the escaping set of an entire function of completely regular growth have positive Lebesgue measure. The essential hypotheses are that the indicator is positive except perhaps at isolated…
The characterization and properties of Julia sets of one parameter family of transcendental meromorphic functions $\zeta_\lambda(z)=\lambda \frac{z}{z+1} e^{-z}$, $\lambda >0$, $z\in \mathbb{C}$ is investigated in the present paper. It is…
Let $g$ and $h$ be transcendental entire functions and let $f$ be a continuous map of the complex plane into itself with $f\circ g=h\circ f.$ Then $g$ and $h$ are said to be semiconjugated by $f$ and $f$ is called a semiconjugacy. We…
We consider the Hausdorff measure of Julia sets and escaping sets of exponential maps with respect to certain gauge functions. We give conditions on the growth of the gauge function which imply that the measure is zero or infinity,…