Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps

We study unimodal interval mapsT with negative Schwarzian derivative satisfying the Collet-Eckmann condition |DTⁿ(Tc)|≧Kλ_cⁿ for some constantsK>0 and λ_c>1 (c is the critical point ofT). We prove exponential mixing properties of the unique invariant probability density ofT, describe the long term behaviour of typical (in the sense of Lebesgue measure) trajectories by Central Limit and Large Deviations Theorems for partial sum processes of the form $$S_n = \Sigma _{i = 0}^{n - 1} f(T^i x)$$ , and study the distribution of "typical" periodic orbits, also in the sense of a Central Limit Theorem and a Large Deviations Theorem. This is achieved by proving quasicompactness of the Perron Frobenius operator and of similar transfer operators for the Markov extension ofT and relating the isolated eigenvalues of these operators to the poles of the corresponding Ruelle zeta functions.