Generalized spectral decomposition and intrinsic irreversibility of the Arnold Cat Map

Abstract We construct a generalized spectral decomposition of the Frobenius-Perron and Koopman operators of the Arnold Cat map. We define a suitable dual pair or rigged Hubert space which provides mathematical meaning to the spectral decomposition. The eigenvalues in the decomposition are the resonances of the power spectrum which determine the decay rates of the correlation functions and the rate of approach to equilibrium. The extended unitary evolution splits into two distinct semigroups which express the intrinsic irreversibility of the Cat map resulting from the strong chaotic properties.

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