Escape rates for the Farey map with approximated holes

We study the escape rate for the Farey map, an infinite measure preserving system, with a hole including the indifferent fixed point. Due to the ergodic properties of the map, the standard theoretical approaches to this problem cannot be applied. It has been recently shown in cite{KM} how to apply the standard analytical methods to a piecewise linear version of the Farey map with holes depending on the associated partition, but their results cannot be obtained in the general case we consider here. To overcome these difficulties we propose here to study approximations of the hole by means of real analytic functions. We introduce a particular family of approximations and study numerically the behavior of the escape rate for approximated holes with vanishing measure. The results suggest that the scaling of the escape rate depends on the ``shape'' of the approximation, and we show that this is a typical feature of systems with an indifferent fixed point, not an artifact of the particular family we consider.