Chaos, External Noise and Fredholm Theory

Abstract

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.