Dynamics of Orthonormal Bases Associated to Basins of Attraction

In the paper "Infinite Product Represenations for Kernels and Iterations of Functions", a technique was developed which allows for the construction of a reproducing kernel Hilbert space on basins of attraction containing $0$. When the right conditions are met, an explicit orthonormal basis can be constructed using a particular class of operators. It is natural then to consider how the orthonormal basis changes as we let the basin of attraction vary. We will consider this question for the basins of attraction containing $0$ of the family of polynomials $\mathcal{F} = \{az^{2^{n+2}}-2az^{2^{n+1}}:a\neq0\}$, where $n\in\mathbb{N}$.