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For non-uniformly expanding maps inducing with a general return time to GibbsMarkov maps, we provide sufficient conditions for obtaining higher orderasymptotics for the correlation function in the infinite measure setting. Alongthe way, we show that these conditions are sufficient to recover previousresults on sharp mixing rates in the finite measure setting for non-Markovmaps, but for a larger class of observables. The results are illustrated by(finite and infinite measure preserving) non-Markov intervals maps with anindifferent fixed point.

Upper and lower bounds for the correlation function via inducing with general return times