The goal of this paper is to accurately describe the maximal zero-free region of the independence polynomial for graphs of bounded degree, for large degree bounds. In previous work with de Boer, Guerini and Regts it was demonstrated that this zero-free region coincides with the normality region of the related occupation ratios. These ratios form a discrete semi-group that is in a certain sense generated by finitely many rational maps. We will show that as the degree bound converges to infinity, the properly rescaled normality regions converge to a limit domain, which can be described as the maximal boundedness component of a semi-group generated by infinitely many exponential maps. We prove that away from the real axis, this boundedness component avoids a neighborhood of the boundary of the limit cardioid, answering a recent question by Andreas Galanis. We also give an exact formula for the boundary of the boundedness component near the positive real boundary point.