renormalization and fixed point fractals

[qooqoo]

http://mathworld.wolfram.com/FixedPoint.html

If the sequence of iterations were treated with a cesaro mean, or another transformation of sequences that preserves limits and maps some divergent sequences to convergent ones, then how might that change the set of fixed points?

A google search hasn't turned up much: "cesaro means and fixed point fractals", "renormalization and fixed point fractals".