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.
D. Alpay, P. Jorgensen, I. Lewkowicz and I. Martziano. Infinite product representations for kernels and iteration of functions. Operator Theory: Advances and Applications, vol. 244 (2015), pp. 67-87.