  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. Then, 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.
