We study infinite products of reproducing kernels with view to their use in dynamics (of iterated function systems), in harmonic analysis, and in stochastic processes. On the way, we construct a new family of representations of the Cuntz relations. T
hen, using these representations we associate a fixed filled Julia set with a Hilbert space. This is based on analysis and conformal geometry of a fixed rational mapping $R$ in one complex variable, and its iterations.
We use reproducing kernel methods to study various rigidity problems. The methods and setting allow us to also consider the non-positive case.
