Absolute Winning Exceptional Sets for Intermittent Interval Maps

We prove that for a Manneville--Pomeau type interval map, the set of points whose orbit closures miss a prescribed countable set is absolute winning in the sense of McMullen.

The proof has three parts.

First we directly prove that the exceptional set for the distinguished endpoint of the induced first-return map is absolute winning.

Then we use the finite-branch winning theorem of Hu--Li--Yu, together with the one-dimensional implication from $1/2$-strong winning to absolute winning, to obtain absolute winning for all countable induced targets.

Finally, a quasisymmetric pullback argument transfers these induced results back to the original map.