In this article, we study about the escaping rate of the Sinai-Ruelle-Bowen (SRB) measure through holes of positive measure constructed in the Julia set of hyperbolic rational maps. An explicit formula for the escaping rate through holes of different sizes is obtained and the dependence of this rate on the position of the hole is discussed. Later, the Hausdorff dimension of the survivor set is also computed. For an easier and better understanding, the simple quadratic polynomial restricted on the unit circle is studied.
"Sinai-Ruelle-Bowen Measure Leaks." Commun. Math. Anal. 13 (1) 1 - 22, 2012.