  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.
