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We study the Lyapunov exponent $L(f,\\mu)$ of $f$ with respect to an $f$-invariant and ergodic Radon probability measure $\\mu$ on the Berkovich Julia set of $f$ and the lower Lyapunov exponent $L_f^{-}(f(c))$ of $f$ at a critical value $f(c)$. Under an integrability assumption, we show $L(f,\\mu)$ has a lower bound only depending on $d$ and $K$. 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We study the Lyapunov exponent $L(f,\\mu)$ of $f$ with respect to an $f$-invariant and ergodic Radon probability measure $\\mu$ on the Berkovich Julia set of $f$ and the lower Lyapunov exponent $L_f^{-}(f(c))$ of $f$ at a critical value $f(c)$. Under an integrability assumption, we show $L(f,\\mu)$ has a lower bound only depending on $d$ and $K$. 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