  Given a critically periodic quadratic map with no secondary
  renormalizations, we introduce the notion of $Q$-recurrent quadratic
  polynomials. We show that the pieces of the principal nest of a
  $Q$-recurrent map $f_c$ converge in shape to the Julia set of $Q$. We use
  this fact to compute analytic invariants of the nest of $f_c$, to give a
  complete characterization of complex quadratic Fibonacci maps and to obtain
  a new auto-similarity result on the Mandelbrot set.
