We compute the spectrum of the transition operator associated to the stochastic adding machine action on ℓ^∞(N), c₀(N) and ℓ¹(N). In particular we show, for example, for the right action on ℓ^∞(N) that the point spectrum is exactly the filled Julia set of some quadratic maps. We also show that the residual spectrum of the action on c₀(N) is a Fatou set of some quadratic maps.