We study the complexity of approximating the value of the independent set polynomial ZG(λ) of a graph G with maximum degree Δ when the activity λ is a complex number. When λ is real, the complexity picture is well-understood, and is captured by two real-valued thresholds λ* and λc, which depend on Δ and satisfy 00, resolving in the affirmative a conjecture of Harvey, Srivastava and Vondrak. Our proof techniques are based around tools from complex analysis — specifically the study of iterative multivariate rational maps
