In this paper we relate the location of the complex zeros of the reliability polynomial to parameters at which a certain family of rational functions derived from the reliability polynomial exhibits chaotic behaviour. We use this connection to prove new results about the location of reliability zeros. In particular we show that there are zeros with modulus larger than $1$ with essentially any possible argument. We moreover use this connection to show that approximately evaluating the reliability polynomial for planar graphs at a non-positive algebraic number in the unit disk is #P-hard.

翻译：本文探讨可靠性多项式复零点分布与一类源自该多项式的有理函数族在特定参数下呈现混沌行为之间的关联。利用这一联系，我们证明了关于可靠性零点分布的新结果。特别地，我们证明了存在模长大于$1$且辐角几乎可为任意值的零点。此外，基于这一关联性，我们证明了在单位圆盘内对平面图的可靠性多项式在非正代数点处进行近似求值是#P难问题。