ランダム反復アルゴリズムのスピングラス理論

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  • Spin-glass Theory of Random Iteration Algorithm

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Cawley and Mauldin analyzed multifractal structure of a probability measure (an invariant measure) ρ induced on a Moran fractal. They introduced a system of weights as well as the probability measure and gave an example in which the multifractal structure presents spin-glass features and showed the gauge invariance. 　 In this paper, we consider the invariant set obtained by a random iteraition algorithm. This random algorithm introduces the assoiated pobability and weight. The multifractal decompositions of the set with respect to the pair of probability density and weight density are considered. To characterize the the Hausdorff dimension of the decomposed sets, we introduce a pair of parameters (q, s). Using these parameters we represent the formula of the Hausdorff dimension. This extension of introducing a pair of parameters gives us the freedom to investigate the spinglass phenomena of mutifractal structure. Furthermore we show the gauge invariance holds.

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