  We introduce a notion of asymptotically orthonormal polynomials for a Borel
measure $\mu$ with compact nonpolar support in $\mathbb{C}$. Such sequences of
polynomials have similar convergence properties of the sequences of Julia sets
and filled Julia sets to those for sequences of orthonormal polynomials. We
give examples of measures for which the monic orthogonal polynomials are
asymptotically orthonormal. Combining this with observations on conformal
invariance of orthogonal polynomials we explore the measure dependency of the
associated dynamics of orthogonal polynomials. Concretely, we study the
dynamics of sequences of asymptotically orthonormal polynomials for the
pullback measure $\phi^\ast(\mu)$ under affine mappings $\phi$. We prove that
the sequences of Julia sets and filled Julia sets of affine deformations of
sequences of asymptotically orthonormal polynomials for $\mu$ also have the
same convergence properties as the Julia sets and filled Julia sets of the
orthonormal polynomials. This leads to theorems on the convergence properties
of affine deformations of the family of iterates of any fixed monic centered
polynomial and, in the case the polynomial is hyperbolic, on the corresponding
family of affine parameter spaces.
