On the escape rate for intermittent maps with holes shrinking around the indifferent fixed point

We study non-uniformly expanding maps of the unit interval with a parabolic fixed point at the origin that admit an ergodic absolutely continuous invariant measure, which may be finite or infinite. By introducing a hole defined by an interval containing the parabolic fixed point, we analyze the escape rate of the resulting open system and its asymptotic behavior as the hole shrinks. Our approach relies on the transfer operator associated with the dynamical system and on the relationship between the transfer operators of the original system and its induced version. The results extend to this general framework previous investigations which considered special cases.