  Area-preserving maps have been observed to undergo a universal
period-doubling cascade, analogous to the famous Feigenbaum-Coullet-Tresser
period doubling cascade in one-dimensional dynamics. A renormalization approach
has been used by Eckmann, Koch and Wittwer in a computer-assisted proof of
existence of a conservative renormalization fixed point. Furthermore, it has
been shown by Gaidashev, Johnson and Martens that infinitely renormalizable
maps in a neighborhood of this fixed point admit invariant Cantor sets with
vanishing Lyapunov exponents on which dynamics for any two maps is smoothly
conjugate. This rigidity is a consequence of an interplay between the decay of
geometry and the convergence rate of renormalization towards the fixed point.
In this paper we prove a result which is crucial for a demonstration of
rigidity: that an upper bound on this convergence rate of renormalizations of
infinitely renormalizable maps is sufficiently small.
