  It has been shown in (Gaidashev et al, 2010) and (Gaidashev et al, 2011) that
infinitely renormalizable area-preserving maps admit invariant Cantor sets with
a maximal Lyapunov exponent equal to zero. Furthermore, the dynamics on these
Cantor sets for any two infinitely renormalizable maps is conjugated by a
transformation that extends to a differentiable function whose derivative is
Holder continuous of exponent alpha>0.
  In this paper we investigate numerically the specific value of alpha. We also
present numerical evidence that the normalized derivative cocycle with the base
dynamics in the Cantor set is ergodic. Finally, we compute renormalization
eigenvalues to a high accuracy to support a conjecture that the renormalization
spectrum is real.
