Absolute winning properties of exceptional sets for rational maps

We study Schmidt’s game on the Julia set $J(f)$ of weak hyperbolic rational maps $f:\text{CC}\to \text{CC}$. For each $y\in J(f)$, let $E(f,y)$ be the set of points in $J(f)$ whose forward orbits stay a positive distance from $y$. We prove that $E(f,y)$ is absolute winning on $J(f)$ for each $y\in J(f)$. To do this, we first prove the absolute winning property for the exceptional set associated to the critical points by a potential game argument, and then use uniform expansion to obtain the result for each target point $y$.