  We give examples of infinitely renormalizable quadratic polynomials $F_c:
z\maps to z^2+c$ with stationary combinatorics whose Julia sets have Hausdorff
dimension arbitrar y close to 1. The combinatorics of the renormalization
involved is close to the Chebyshev one . The argument is based upon a new tool,
a ``Recursive Quadratic Estimate'' for the Poincar\'e series of an infinitely
re normalizable map.
