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We investigate the distribution of $\\lambda_{\\alpha}= \\sum_{i=0}^{t-1} \\ln \\left| M'(x_i) \\right|/t^{\\alpha}$, where $\\alpha$ is determined by the nonlinearity of the map in the vicinity of marginally unstable fixed points. The mean of $\\lambda_{\\alpha}$ is determined by the infinite invariant density. 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