# Asymptotic escape rates and limiting distributions for multimodal maps

## TL;DR

This paper investigates escape rates and limiting distributions in multimodal maps with holes, establishing a variational principle and scaling limits for small holes, applicable to both periodic and nonperiodic points.

## Contribution

It introduces a comprehensive analysis of escape dynamics in multimodal maps with holes, including a variational principle and scaling limits without restrictive hole placement conditions.

## Key findings

- Escape rates are uniform for a large class of initial distributions.
- A variational principle links escape rate to pressure on the survivor set.
- Scaling limits for escape rates are established as holes shrink to points.

## Abstract

We consider multimodal maps with holes and study the evolution of the open systems with respect to equilibrium states for both geometric and H\"older potentials. For small holes, we show that a large class of initial distributions share the same escape rate and converge to a unique absolutely continuous conditionally invariant measure; we also prove a variational principle connecting the escape rate to the pressure on the survivor set, with no conditions on the placement of the hole. Finally, introducing a weak condition on the centre of the hole, we prove scaling limits for the escape rate for holes centred at both periodic and nonperiodic points, as the diameter of the hole goes to zero.