  Let f(z)=z^2+c be an infinitely renormalizable quadratic polynomial and
J_\infty be the intersection of forward orbits of "small" Julia sets of its
simple renormalizations. We prove that if f admits an infinite sequence of
satellite renormalizations, then every invariant measure of f: J_\infty\to
J_\infty is supported on the postcritical set and has zero Lyapunov exponent.
Coupled with [G. Levin, F. Przytycki, W. Shen, The Lyapunov exponent of
holomorphic maps. Invent. Math. 205 (2016), 363-382], this implies that the
Lyapunov exponent of such f at c is equal to zero, which answers partly a
question posed by Weixiao Shen.
