Related papers: Sur une question de Bergweiler
We study how the orbits of the singularities of the inverse of a meromorphic function prescribe the dynamics on its Julia set, at least up to a set of (Lebesgue) measure zero. We concentrate on a family of entire transcendental functions…
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…
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…
Nevanlinna functions are meromorphic functions with a finite number of asymptotic values and no critical values. In [KK2] it was proved that if the orbits of all the asymptotic values accumulate on a compact set on which the function acts…
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…
We survey the definition of the radial Julia set of a meromorphic function (in fact, more generally, any "Ahlfors islands map"), and give a simple proof that the Hausdorff dimension of the reduced Julia set always coincides with the…
We present a p-adic and non-archimdean version of the Five Islands Theorem for meromorphic functions from Ahlfors' theory of covering surfaces. In the non-archimedean setting, the theorem requires only four islands, with explicit constants.…
It is an open problem whether repelling periodic points are dense in the classical Julia set of a non-archimedean rational function of degree more than one. We give a partial positive answer to this question based on a study of a…
Let $f$ and $g$ be commuting meromorphic functions with finitely many poles. By studying the behaviour of Fatou components under this commuting relation, we prove that $f$ and $g$ have the same Julia set whenever $f$ and $g$ have no simply…
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.…
A tropical version of Nevanlinna theory is described in which the role of meromorphic functions is played by continuous piecewise linear functions of a real variable whose one-sided derivatives are integers at every point. These functions…
We discuss the dynamics of semigroups of transcendental entire functions using Fatou-Julia theory and provide a condition for the complete invariance of escaping set and Julia set of transcendental semigroups. Results regarding limit…
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…
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…
We study transcendental meromorphic functions having two prepole asymptotic values and no critical values. We prove that these functions acting on their Julia sets are non-ergodic, which illustrates the antithesis of the Keen-Kotus result…
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 prove that if the multipliers of the repelling periodic orbits of a complex polynomial grow at least like $n^{5 + \epsilon}$, for some $\epsilon > 0$, then the Julia set of the polynomial is locally connected when it is connected. As a…
In this paper we study two classes of meromorphic functions previously studied by Mayer, Kotus, and Urba\'nski. In particular we estimate a lower bound for the Julia set and the set of escaping points for non-autonomous additive and affine…
We will apply Nevanlinna Theory to prove several Ax-Schanuel type Theorems for functional transcendence when the exponential map is replaced by other meromorphic functions. We also show that analytic dependence will imply algebraic…
We study transcendental singularities of a Schr\"oder map arising from a rational function $f$, using results from complex dynamics and Nevanlinna theory. These maps are transcendental meromorphic functions of finite order in the complex…