One of the fundamental questions in the study of dynamical systems is how quickly does a dynamical system lose mass when we drill a small hole in its phase space. This quantity is termed as ‘Escape Rate’. For systems with uniform expansion, the escape rate into a hole behaves linearly with the measure of the hole as it shrinks to a point. The situation becomes much more subtle when the system has good expansion everywhere else except at an indifferent fixed point.

In this talk, we discuss dynamics of intermittent maps of the unit interval having one indifferent fixed point at $0$. A prototypical example of such a system is the Pomeau–Manneville map. The map forces orbits to spend rather a large amount of time in neighbourhood of $0$. Suppose a hole is drilled in this system that contains the indifferent point. We observe that the escape rate does not always scale linearly with the size of the hole as it shrinks to $0$. Instead, as a consequence of the slow dynamics near $0$, different scaling laws are observed depending on the strength of intermittency.

This is a joint work with Claudio Bonanno.