Decay of correlations for critically intermittent systems

Abstract

For a family of random intermittent dynamical systems with a superattracting fixed point we prove that a phase transition occurs for the existence of an absolutely continuous invariant probability measure depending on the randomness parameters and the orders of the maps at the superattracting fixed point. In case the systems have an absolutely continuous invariant probability measure, we show that the systems are mixing and that correlations decay polynomially even though some of the deterministic maps present in the system have exponential decay of correlations. This contrasts other known results, where a system maintains exponential decay of correlations under stochastic perturbations of a deterministic map with exponential rate of mixing, see e.g. Baladi and Viana (1996 Ann. Sci. l’Ecole Norm. Sup. 29 483–517).