On the connectivity of the Julia sets of meromorphic functions

We prove that every transcendental meromorphic map f with a disconnected Julia set has a weakly repelling fixed point. This implies that the Julia set of Newton's method for finding zeroes of an entire map is connected. Moreover, extending a result of Cowen for holomorphic self-maps of the disc, we show the existence of absorbing domains for holomorphic self-maps of hyperbolic regions whose iterates tend to a boundary point. In particular, the results imply that periodic Baker domains of Newton's method for entire maps are simply connected, which solves a well-known open question.Comment: 34 pages, 10 figure

Accesses to infinity from Fatou components

Agraïments: Supported by the Polish NCN grant decision DEC-2012/06/M/ST1/00168.We study the boundary behaviour of a meromorphic map f: \C C on its invariant simply connected Fatou component U. To this aim, we develop the theory of accesses to boundary points of U and their relation to the dynamics of f. In particular, we establish a correspondence between invariant accesses from U to infinity or weakly repelling points of f and boundary fixed points of the associated inner function on the unit disc. We apply our results to describe the accesses to infinity from invariant Fatou components of the Newton maps

Connectivity of Julia sets of Newton maps: a unified approach

In this paper we present a unified proof of the fact that the Julia set of Newton's method applied to a holomorphic function on the complex plane (a polynomial of degree larger than $1$ or a transcendental entire function) is connected. The result was recently completed by the authors' previous work, as a consequence of a more general theorem whose proof spreads among many papers, which consider separately a number of particular cases for rational and transcendental maps, and use a variety of techniques. In this note we present a unified, direct and reasonably self-contained proof which works in all situations alike

Accesses to infinity from Fatou components.

We study the boundary behaviour of a meromorphic map $f\mathbb{C} \rightarrow \widehat{C}$ on its invariant simply connected Fatou component $U$. To this aim, we develop the theory of accesses to boundary points of $U$ and their relation to the dynamics of $f$. In particular, we establish a correspondence between invariant accesses from $U$ to infinity or weakly repelling points of $f$ and boundary fixed points of the associated inner function on the unit disc. We apply our results to describe the accesses to infinity from invariant Fatou components of the Newton maps

Absorbing sets and Baker domains for holomorphic maps

We consider holomorphic maps $f: U \rightarrow U$ for a hyperbolic domain $U$ in the complex plane, such that the iterates of $f$ converge to a boundary point $\zeta$ of $U$. By a previous result of the authors, for such maps there exist nice absorbing domains $W \subset U$. In this paper we show that $W$ can be chosen to be simply connected, if $f$ has doubly parabolic type in the sense of the Baker-Pommerenke-Cowen classification of its lift by a universal covering (and $\zeta$ is not an isolated boundary point of $U$). We also provide counterexamples for other types of the map $f$ and give an exact characterization of doubly parabolic type in terms of the dynamical behaviour of $f$

Escaping points in the boundaries of baker domains

We study the dynamical behaviour of points in the boundaries of simply connected invariant Baker domains $U$ of meromorphic maps $f$ with a finite degree on $U$. We prove that if $f|_U$ is of hyperbolic or simply parabolic type, then almost every point in the boundary of $U$, with respect to harmonic measure, escapes to infinity under iteration of $f$. On the contrary, if $f|_U$ is of doubly parabolic type, then almost every point in the boundary of $U$, with respect to harmonic measure, has dense forward trajectory in the boundary of $U$, in particular the set of escaping points in the boundary of $U$ has harmonic measure zero. We also present some extensions of the results to the case when $f$ has infinite degree on $U$, including classical Fatou example

Thermodynamic formalism for meromorphic maps

I will describe recent results concerning the existence of some elements of thermodynamic formalism, such as topological pressure, Bowen's formula and conformal measures, for a large class of transcendental entire and meromorphic maps on the complex plane. This is a joint work with Bogusława Karpińska and Anna Zdunik.Non UBCUnreviewedAuthor affiliation: University of WarsawFacult

From Newton's method to exotic basins - Part II: Bifurcation of the Mandelbrot-like sets

This is a continuation of the work [Ba] dealing with the family of all cubic rational maps with two supersinks. We prove the existence of a parabolic bifurcation of the Mandelbrot-like sets in the parameter space. Starting from a Mandelbrot-like set in cubic Newton maps and changing parameters in a continuous way, we obtain a parabolic Mandelbrot-like set contained in the family of maps with a fixed point of multiplier 1. Then it bifurcates to two paths of Mandelbrot-like sets -- one contained in the set of maps with exotic basins and the other in the set of maps with non-exotic basins. The non-exotic path ends at a Mandelbrot-like set in cubic polynomials. 1 Introduction This paper extends the work started in [Ba]. We study the family F = ff a;b g, where f a;b (z) = az 2 bz + 1 \Gamma 2b (2 \Gamma b)z \Gamma 1 ; a 2 C n f0g; b 2 C n f0; 1g: (1) This family consists of cubic rational maps with two supersinks: 0 and 1 and a critical point 1. As one can easily check, the critical p..