We consider multimodal maps with holes and study the evolution of the open systems with respect to equilibrium states for both geometric and Hölder 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 non-periodic points, as the diameter of the hole goes to zero.

中文翻译：

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多模式地图的渐近逃逸率和限制分布

我们考虑带孔的多峰映射，并研究开放系统相对于几何和 Hölder 势的平衡状态的演变。对于小洞，我们展示了一大类初始分布共享相同的逃逸率并收敛到唯一的绝对连续条件不变测度；我们还证明了一个将逃逸率与幸存者组上的压力联系起来的变分原理，对孔的位置没有任何条件。最后，在孔的中心引入一个弱条件，我们证明了以周期性和非周期性点为中心的孔的逃逸率的缩放限制，因为孔的直径变为零。