I am looking for an example of a function with a Julia set of positive Lebesgue measure. I am certain these exist but cannot find a single example in the papers I have looked at. The function does not have to be a quadratic, it can be of any form.
Thank you.
There are many known classes of rational functions whose Julia set is the entire Riemann sphere. For example, it is known that a post-critically finite rational function with no periodic critical points has the entire Riemann sphere as its Julia set. (This is Corollary 16.5 of Milnor's Dynamics in One Complex Variable.) Perhaps the simplest such example is the rational function $$ f(z)=1-2z^{-2}. $$ This has critical points $\{0,\infty\}$, with $f(0)=\infty$, $f(\infty)=1$, $f(1)=-1$, and $f(-1)=-1$.
Other Julia sets with positive Lebesgue measure are known. For example, Curt McMullen proved in 1987 that any function of the form $f(z) = \sin(\alpha z+\beta)$ (where $\alpha\ne 0$) or the form $f(z)=\gamma e^z + \delta e^{-z}$ (with $\gamma,\delta\ne 0$) has a Julia set with positive Lebesgue measure. See:
C. McMullen, Area and Hausdorff dimension of Julia sets of entire functions. Transactions of the American Mathematical Society 300.1 (1987): 329-342. Crossref.
In 2005, Xavier Buff and Arnaud Chéritat proved that there exist quadratic polynomials whose Julia sets have positive Lebesgue measure. See:
X. Buff and A. Chéritat, Quadratic Julia sets with positive area. Annals of Mathematics (2012): 673-746. JSTOR. arXiv.