Abstract:

Let $f$ and $g$ two holomorphic functions (either both rational or entire). Assume they commute, that is $fcirc g= gcirc f$. It is known since Julia and Fatou that in the rational case we have $J(f)=J(g)$. The entire transcendental case is still open (so we do not know if this is true or not). After some partials results due to I. Baker, W. Bergweiler showed that if the fast escaping set is in the Julia set (for both maps) then $J(f)=J(g)$. In particular this implies that if $f$ has no (fast) escaping wandering domains then $J(f)=J(g)$. Recently, Anna Miriam Benini, Phil Rippon and Gwyneth Stallard improved this statement by showing that if $f$ and $g$ has no simply connected (fast) escaping wandering domains then $J(f)=J(g)$. I will present, and partially discuss, those results.