Correlation Spectrum of Quenched and Annealed Equilibrium States for Random Expanding Maps

We show that the integrated transfer operators for positively weighted independent identically distributed smooth expanding systems give rise to annealed equilibrium states for a new variational principle. The unique annealed equilibrium state coincides with the unique annealed Gibbs state. Using work of Ruelle 1990] and Fried 1995] on generalised Fredholm determinants for transfer operators, we prove that the discrete spectrum of the transfer operators coincides with the correlation spectrum of these invariant measures (yielding exponential decay of correlations), and with the poles of an annealed zeta function, defined also for complex weights. A modified integrated transfer operator is introduced, which describes the (relativised) quenched states studied e.g. by Kifer 1992], and conditions (including SRB) ensuring coincidence of quenched and annealed states are given. For small random perturbations we obtain stability results on the quenched and annealed measures and spectra by applying perturbative results of Young and the author 1993]. Received: 16 April 1996 / Accepted: 25 October 1996