arXiv subjects

Xavier Jarque

Publications and source records attributed to Xavier Jarque.

22 records · Page 2Linked to original sources

arXiv2012

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.

math.DS↗

arXiv2012

Brushing the hairs of transcendental entire functions

Let f be a hyperbolic transcendental entire function of finite order in the Eremenko-Lyubich class (or a finite composition of such maps), and suppose that f has a unique Fatou component. We show that the Julia set of $f$ is a Cantor bouquet; i.e. is ambiently homeomorphic to a straight brush in the sense of Aarts and Oversteegen. In particular, we show that any two such Julia sets are ambiently homeomorphic. We also show that if $f\in\B$ has finite order (or is a finite composition of such maps), but is not necessarily hyperbolic, then the Julia set of f contains a Cantor bouquet. As part of our proof, we describe, for an arbitrary function $f\in\B$, a natural compactification of the dynamical plane by adding a "circle of addresses" at infinity.

math.DS↗

arXiv2011

Non-landing hairs in Sierpinski curve Julia sets of transcendental entire maps

We consider the family of transcendental entire maps given by $f_a(z)=a(z-(1-a))\exp(z+a)$ where $a$ is a complex parameter. Every map has a superattracting fixed point at $z=-a$ and an asymptotic value at $z=0$. For $a>1$ the Julia set of $f_a$ is known to be homeomorphic to the Sierpiński universal curve, thus containing embedded copies of any one-dimensional plane continuum. In this paper we study subcontinua of the Julia set that can be defined in a combinatorial manner. In particular, we show the existence of non-landing hairs with prescribed combinatorics embedded in the Julia set for all parameters $a\geq 3$. We also study the relation between non-landing hairs and the immediate basin of attraction of $z=-a$. Even as each non-landing hair accumulates onto the boundary of the immediate basin at a single point, its closure, nonetheless, becomes an indecomposable subcontinuum of the Julia set.

math.DS↗

arXiv2010

On the connectivity of the escaping set for complex exponential Misiurewicz parameters

Let $E_{\la}(z)=\la {\rm exp}(z), \ λ\in \mathbb C$ be the complex exponential family. For all functions in the family there is a unique asymptotic value at 0 (and no critical values). For a fixed $\la$, the set of points in $\mathbb C$ with orbit tending to infinity is called the escaping set. We prove that the escaping set of $E_{\la}$ with $\la$ Misiurewicz (that is, a parameter for which the orbit of the singular value is strictly preperiodic) is a connected set.

math.DS↗