Recently, jointly with Scott Sutherland we showed that the Feigenbaum map has Julia set
of Hausdorff dimension less than two. In this talk I will present a generalization of our approach
to real periodic points of Feigenbaum renormalization.
I will give a sufficient condition for the Julia set of such map to have Hausdorff dimension less than two.
This condition can be verified numerically. Empirical computations suggest that for low periods it is satisfied,
and so the corresponding Julia sets have Hausdorff dimension less than two.