  The independence polynomial originates in statistical physics as the
partition function of the hard-core model. The location of the complex zeros of
the polynomial is related to phase transitions, and plays an important role in
the design of efficient algorithms to approximately compute evaluations of the
polynomial.
  In this paper we directly relate the location of the complex zeros of the
independence polynomial to computational hardness of approximating evaluations
of the independence polynomial. We do this by moreover relating the location of
zeros to chaotic behaviour of a naturally associated family of rational
functions; the occupation ratios.
