On invariant measures of 'satellite' infinitely renormalizable quadratic polynomials

Let f (z) = z2 + c be an infinitely renormalizable quadratic polynomial and J∞ 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∞ → J∞ is supported on the postcritical set and has zero Lyapunov exponent. Coupled with, this implies that the Lyapunov exponent of such f at c is equal to zero, which partly answers a question posed by Weixiao Shen.