Strong-Winning Target Avoidance for Manneville--Pomeau Maps

Abstract

We prove that target-avoidance sets for Manneville--Pomeau maps are strong winning for Schmidt's game. More precisely, for the class of nonuniformly expanding interval maps considered here, there exists a single parameter $α>0$ such that for every target $p\in[0,1]$, the set of points whose forward orbit does not accumulate on $p$ is $α$-strong winning. The proof induces on the uniformly expanding region $[r_1,1]$. The resulting first-return map has infinitely many branches, so we approximate it by finite-branch expanding maps, apply a theorem of Hu--Li--Yu to those finite approximants, and then transfer the resulting strategies first to the induced map and then to the original Manneville--Pomeau map.