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The Fredholm theory of integral equations is applied to the Perron-Frobenius equation which determines the invariant measure of nonlinear difference equations. The resultant Fredholm determinant D is identical to 1/ζ. where ζ is the Artine-Mazure-Ruelle ζ-function. From the investigation of D we get the observable condition of the invariant measure of chaos, its stability against external noise, etc.
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Oono et al. (1980) studied this question.
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