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Motivated by recent studies on the fractal nature of folded proteins, we analyze the time-dependent autocorrelation function of the distance between two points on a thermally vibrating fractal. Using fractons, the vibrational excitations of a fractal, we show that for both strongly underdamped and overdamped vibrations this correlation function decays anomalously, displaying a crossover from a nearly stretched exponential decay at short times to a slow algebraic decay at long times. Relationship to single molecule experiments is discussed.