Transcendental Julia sets of positive finite Lebesgue measure
We show that there exists a transcendental entire function whose Julia set has positive finite Lebesgue measure.
math.DS↗
arXiv subjects
Mareike Wolff
Publications and source records attributed to Mareike Wolff.
A class of Newton maps with Julia sets of Lebesgue measure zero
Let $g(z)=\int_0^zp(t)\exp(q(t))\,dt+c$ where $p,q$ are polynomials and $c\in\mathbb{C}$, and let $f$ be the function from Newton's method for $g$. We show that under suitable assumptions the Julia set of $f$ has Lebesgue measure zero. Together with a theorem by Bergweiler, our result implies that $f^n(z)$ converges to zeros of $g$ almost everywhere in $\mathbb{C}$ if this is the case for each zero of $g''$. In order to prove our result, we establish general conditions ensuring that Julia sets have Lebesgue measure zero.
math.DS↗
Exponential polynomials with Fatou and non-escaping sets of finite Lebesgue measure
We give conditions ensuring that the Fatou set and the complement of the fast escaping set of an exponential polynomial $f$ have finite Lebesgue measure. Essentially, these conditions are designed such that $|f(z)|\ge\exp(|z|^α)$ for some $α>0$ and all $z$ outside a set of finite Lebesgue measure.
math.DS↗