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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 [13], this implies that the Lyapunov exponent of such f at c is equal to zero, which partly answers a question posed by Weixiao Shen.
